Proof of fully faithfulness. Since vectorbundles on $Y$ are locally trivial, this reduces to the statement that $$ \colim_F \Gamma(Y, \mathcal{O}_Y) \longrightarrow \colim_F \Gamma(X, \mathcal{O}_X) $$ is bijective. Since $\{X \to Y\}$ is an h covering, this will follow from Lemma \ref{lemma-h-sheaf-colim-F} if we can show that the two maps $$ \colim_F \Gamma(X, \mathcal{O}_X) \longrightarrow \colim_F \Gamma(X \times_Y X, \mathcal{O}_{X \times_Y X}) $$ are equal. Let $g \in \Gamma(X, \mathcal{O}_X)$ and denote $g_1$ and $g_2$ the two pullbacks of $g$ to $X \times_Y X$. Since $X_{y, red}$ is geometrically connected, we see that $H^0(X_{y, red}, \mathcal{O}_{X_{y, red}})$ is a purely inseparable extension of $\kappa(y)$, see Varieties, Lemma \ref{varieties-lemma-proper-geometrically-reduced-global-sections}. Thus $g^q|_{X_{y, red}}$ comes from an element of $\kappa(y)$ for some $p$-power $q$ (which may depend on $y$). It follows that $g_1^q$ and $g_2^q$ map to the same element of the residue field at any point of $(X \times_Y X)_y = X_y \times_y X_y$. Hence $g_1 - g_2$ restricts to zero on $(X \times_Y X)_{red}$. Hence $(g_1 - g_2)^n = 0$ for some $n$ which we may take to be a $p$-power as desired. \medskip\noindent Description of essential image. Let $\mathcal{E}$ be as in the statement of the proposition. We first reduce to the Noetherian case. \medskip\noindent Let $y \in Y$ be a point and view it as a morphism $y \to Y$ from the spectrum of the residue field into $Y$. We can write $y \to Y$ as a filtered limit of morphisms $Y_i \to Y$ of finite presentation with $Y_i$ affine. (It is best to prove this yourself, but it also follows formally from Limits, Lemma \ref{limits-lemma-relative-approximation} and \ref{limits-lemma-limit-affine}.) For each $i$ set $Z_i = Y_i \times_Y X$. Then $X_y = \lim Z_i$ and $X_{y, red} = \lim Z_{i, red}$. By Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation} we can find an $i$ such that $F^{n_y, *}\mathcal{E}|_{Z_{i, red}} \cong \mathcal{O}_{Z_{i, red}}^{\oplus r_y}$. Fix $i$. We have $Z_{i, red} = \lim Z_{i, j}$ where $Z_{i, j} \to Z_i$ is a thickening of finite presentation (Limits, Lemma \ref{limits-lemma-closed-is-limit-closed-and-finite-presentation}). Using the same lemma as before we can find a $j$ such that $F^{n_y, *}\mathcal{E}|_{Z_{i, j}} \cong \mathcal{O}_{Z_{i, j}}^{\oplus r_y}$. We conclude that for each $y \in Y$ there exists a morphism $Y_y \to Y$ of finite presentation whose image contains $y$ and a thickening $Z_y \to Y_y \times_Y X$ such that $F^{n_y, *}\mathcal{E}|_{Z_y} \cong \mathcal{O}_{Z_y}^{\oplus r_y}$. Observe that the image of $Y_y \to Y$ is constructible (Morphisms, Lemma \ref{morphisms-lemma-chevalley}). Since $Y$ is quasi-compact in the constructible topology (Topology, Lemma \ref{topology-lemma-constructible-hausdorff-quasi-compact} and Properties, Lemma \ref{properties-lemma-quasi-compact-quasi-separated-spectral}) we conclude that there are a finite number of morphisms $$ Y_1 \to Y,\ Y_2 \to Y,\ \ldots,\ Y_N \to Y $$ of finite presentation such that $Y = \bigcup \Im(Y_a \to Y)$ set theoretically and such that for each $a \in \{1, \ldots, N\}$ there exist integers $n_a, r_a \geq 0$ and there is a thickening $Z_a \subset Y_a \times_Y X$ of finite presentation such that $F^{n_a, *}\mathcal{E}|_{Z_a} \cong \mathcal{O}_{Z_a}^{\oplus r_a}$. \medskip\noindent Formulated in this way, the condition descends to an absolute Noetherian approximation. We stronly urge the reader to skip this paragraph. First write $Y = \lim_{i \in I} Y_i$ as a cofiltered limit of schemes of finite type over $\mathbf{F}_p$ with affine transition morphisms (Limits, Lemma \ref{limits-lemma-relative-approximation}). Next, we can assume we have proper morphisms $f_i : X_i \to Y_i$ whose base change to $Y$ recovers $f : X \to Y$, see Limits, Lemma \ref{limits-lemma-descend-finite-presentation}. After increasing $i$ we may assume there exists a finite locally free $\mathcal{O}_{X_i}$-module $\mathcal{E}_i$ whose pullback to $X$ is isomorphic to $\mathcal{E}$, see Limits, Lemma \ref{limits-lemma-descend-invertible-modules}. Pick $0 \in I$ and denote $E \subset Y_0$ the constructible subset where the geometric fibres of $f_0$ are connected, see More on Morphisms, Lemma \ref{more-morphisms-lemma-nr-geom-connected-components-constructible}. Then $Y \to Y_0$ maps into $E$, see More on Morphisms, Lemma \ref{more-morphisms-lemma-base-change-fibres-geometrically-connected}. Thus $Y_i \to Y_0$ maps into $E$ for $i \gg 0$, see Limits, Lemma \ref{limits-lemma-limit-contained-in-constructible}. Hence we see that the fibres of $f_i$ are geometrically connected for $i \gg 0$. By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} for large enough $i$ we can find morphisms $Y_{i, a} \to Y_i$ of finite type whose base change to $Y$ recovers $Y_a \to Y$, $a  \in \{1, \ldots, N\}$. After possibly increasing $i$ we can find thickenings $Z_{i, a} \subset Y_{i, a} \times_{Y_i} X_i$ whose base change to $Y_a \times_Y X$ recovers $Z_a$ (same reference as before combined with Limits, Lemmas \ref{limits-lemma-descend-closed-immersion-finite-presentation} and \ref{limits-lemma-descend-surjective}). Since $Z_a = \lim Z_{i, a}$ we find that after increasing $i$ we may assume $F^{n_a, *}\mathcal{E}_i|_{Z_{i, a}} \cong \mathcal{O}_{Z_{i, a}}^{\oplus r_a}$, see Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation}. Finally, after increasing $i$ one more time we may assume $\coprod Y_{i, a} \to Y_i$ is surjective by Limits, Lemma \ref{limits-lemma-descend-surjective}. At this point all the assumptions hold for $X_i \to Y_i$ and $\mathcal{E}_i$ and we see that it suffices to prove result for $X_i \to Y_i$ and $\mathcal{E}_i$. \medskip\noindent Assume $Y$ is of finite type over $\mathbf{F}_p$. To prove the result we will use induction on $\dim(Y)$. We are trying to find an object of $\colim_F \textit{Vect}(Y)$ which pulls back to the object of $\colim_F \textit{Vect}(X)$ determined by $\mathcal{E}$. By the fully faithfulness already proven and because of Proposition \ref{proposition-h-descent-vector-bundles-p} it suffices to construct a descent of $\mathcal{E}$ after replacing $Y$ by the members of a h covering and $X$ by the corresponding base change. This means that we may replace $Y$ by a scheme proper and surjective over $Y$ provided this does not increase the dimension of $Y$. If $T \subset T'$ is a thickening of schemes of finite type over $\mathbf{F}_p$ then $\colim_F \textit{Vect}(T) = \colim_F \textit{Vect}(T')$ as $\{T \to T'\}$ is a h covering such that $T \times_{T'} T = T$. If $T' \to T$ is a universal homeomorphism of schemes of finite type over $\mathbf{F}_p$, then $\colim_F \textit{Vect}(T) = \colim_F \textit{Vect}(T')$ as $\{T \to T'\}$ is a h covering such that the diagonal $T \subset T \times_{T'} T$ is a thickening. \medskip\noindent Using the general remarks made above, we may and do replace $X$ by its reduction and we may assume $X$ is reduced. Consider the Stein factorization $X \to Y' \to Y$, see More on Morphisms, Theorem \ref{more-morphisms-theorem-stein-factorization-Noetherian}. Then $Y' \to Y$ is a universal homeomorphism of schemes of finite type over $\mathbf{F}_p$. By the above we may replace $Y$ by $Y'$. Thus we may assume $f_*\mathcal{O}_X = \mathcal{O}_Y$ and that $Y$ is reduced. This reduces us to the case discussed in the next paragraph. \medskip\noindent Assume $Y$ is reduced and $f_*\mathcal{O}_X = \mathcal{O}_Y$ over a dense open subscheme of $Y$. Then $X \to Y$ is flat over a dense open subscheme $V \subset Y$, see Morphisms, Proposition \ref{morphisms-proposition-generic-flatness-reduced}. By Lemma \ref{lemma-flat-after-blowing-up} there is a $V$-admissible blowing up $Y' \to Y$ such that the strict transform $X'$ of $X$ is flat over $Y'$. Observe that $\dim(Y') = \dim(Y)$ as $Y$ and $Y'$ have a common dense open subscheme. By More on Morphisms, Lemma \ref{more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous} and the fact that $V \subset Y'$ is dense all fibres of $f' : X' \to Y'$ are geometrically connected. We still have $(f'_*\mathcal{O}_{X'})|_V = \mathcal{O}_V$. Write $$ Y' \times_Y X = X' \cup E \times_Y X $$ where $E \subset Y'$ is the exceptional divisor of the blowing up. By the general remarks above, it suffices to prove existence for $Y' \times_Y X \to Y'$ and the restriction of $\mathcal{E}$ to $Y' \times_Y X$. Suppose that we find some object $\xi'$ in $\colim_F \textit{Vect}(Y')$ pulling back to the restriction of $\mathcal{E}$ to $X'$ (viewed as an object of the colimit category). By induction on $\dim(Y)$ we can find an object $\xi''$ in $\colim_F \textit{Vect}(E)$ pulling back to the restriction of $\mathcal{E}$ to $E \times_Y X$. Then the fully faithfullness determines a unique isomorphism $\xi'|_E \to \xi''$ compatible with the given identifications with the restriction of $\mathcal{E}$ to $E \times_{Y'} X'$. Since $$ \{E \times_Y X \to Y' \times_Y X, X' \to Y' \times_Y X\} $$ is a h covering given by a pair of closed immersions with $$ (E \times_Y X) \times_{(Y' \times_Y X)} X' = E \times_{Y'} X' $$ we conclude that $\xi'$ pulls back to the restriction of $\mathcal{E}$ to $Y' \times_Y X$. Thus it suffices to find $\xi'$ and we reduce to the case discussed in the next paragraph. \medskip\noindent Assume $Y$ is reduced, $f$ is flat, and $f_*\mathcal{O}_X = \mathcal{O}_Y$ over a dense open subscheme of $Y$. In this case we consider the normalization $Y^\nu \to Y$ (Morphisms, Section \ref{morphisms-section-normalization}). This is a finite surjective morphism (Morphisms, Lemma \ref{morphisms-lemma-nagata-normalization} and \ref{morphisms-lemma-ubiquity-nagata}) which is an isomorphism over a dense open. Hence by our general remarks we may replace $Y$ by $Y^\nu$ and $X$ by $Y^\nu \times_Y X$. After this replacement we see that $\mathcal{O}_Y = f_*\mathcal{O}_X$ (because the Stein factorization has to be an isomorphism in this case; small detail omitted). \medskip\noindent Assume $Y$ is a normal Noetherian scheme, that $f$ is flat, and that $f_*\mathcal{O}_X = \mathcal{O}_Y$. After replacing $\mathcal{E}$ by a suitable Frobenius power pullback, we may assume $\mathcal{E}$ is trivial on the scheme theoretic fibres of $f$ at the generic points of the irreducible components of $Y$ (because $\colim_F \textit{Vect}(-)$ is an equivalence on universal homeomorphisms, see above). Similarly to the arguments above (in the reduction to the Noetherian case) we conclude there is a dense open subscheme $V \subset Y$ such that $\mathcal{E}|_{f^{-1}(V)}$ is free. Let $Z \subset Y$ be a closed subscheme such that $Y = V \amalg Z$ set theoretically. Let $z_1, \ldots, z_t \in Z$ be the generic points of the irreducible components of $Z$ of codimension $1$. Then $A_i = \mathcal{O}_{Y, z_i}$ is a discrete valuation ring. Let $n_i$ be the integer found in Lemma \ref{lemma-trivial-fibres-dvr} for the scheme $X_{A_i}$ over $A_i$. After replacing $\mathcal{E}$ by a suitable Frobenius power pullback, we may assume $\mathcal{E}$ is free over $X_{A_i/\mathfrak m_i^{n_i}}$ (again because the colimit category is invariant under universal homeomorphisms, see above). Then Lemma \ref{lemma-trivial-fibres-dvr} tells us that $\mathcal{E}$ is free on $X_{A_i}$. Thus finally we conclude by applying Lemma \ref{lemma-trivial-over-dvrs}.