> It's impossible to not move in space due to inflation.

Do you mean cosmic inflation? If that happened at all, it must have been before the formation of the cosmic microwave background. The (standard) concordance cosmology provides a calculation for the PVF photon visibility function (PVF) -- when the surface of last scattering became transparent to photons. From detailed observation of the CMB (by the Wilkinson Microwave Anisotropy Probe among many others), we have data strongly supporting that the PVF's interval from opacity to transparency is about 110 000 years, opaque at the early time of about 370 000 years after the electroweak epoch. (The splitting of electroweak into electromagnetism and the weak force gave rise to electrons and other leptons, photons, and neutrinos, and their respective antiparticles; consequently there is also a Cosmic Neutrino Background). Prior to the start of the PVF, matter in the universe was too hot to form electrically neutral structures like atoms, and prior to the end of the PVF these structures would be broken apart by electromagnetic interactions.

Or do you mean the metric expansion of space? That's an ongoing observable, unlike cosmic inflation, which ended hundreds of millions of years before the formation of the first galaxy clusters, while the metric expansion continues to cause all galaxy clusters to drift apart from one another.

"Cosmic inflation" doesn't do anything to the motion of an object today; it switched off more than thirteen billion years ago.

How about expansion, then?

The universe at scales where galaxy clusters are like fine grains of dust or microscopic elements of a fluid is well represented by a set of equations -- the Friedmann equations -- that describe an expanding spacetime (the Robertson-Walker metric (R-W), if we subtract out all the galaxy clusters leaving only vacuum behind). However, the R-W metric is not a good description for galaxy clusters themselves, nor individual galaxies, nor individual stars, etc. Those are best described by a collapsing spacetime, with a metric like Lemaître-Tolman-Bondi (LTB), adapted for hierarchy and non-spherically symmetrical lumpiness of the collapsing matter. (You are on a lump right now! There is obviously a lot of dense mass in one direction, below you, but not so much above you). We can combine R-W and LTB into a "swiss-cheese" model, where the name is evocative of holes (the LTB collapsing spacetimes) embedded in the otherwise smooth, homogeneous, isotropic Friedmann-[Lemaître]-Robertson-Walker spacetime).

Our galaxy is in a "hole", and so there is no metric expansion within our galaxy.

(Or alternatively, and commonly put forward in popsci descriptions of dark energy, the expansion is so small within our solar system that we can ignore it. We have checked for local expansion experimentally, because if we could measure local expansion we might choose to explore otherwise-superfluous theoretical ideas. All measurements so far are consistent with no expansion in our solar system.)

Is it impossible to not move in space, as you say? I don't know. One can prove whether one is in gravitational free-fall, using highly sensitive accelerometers. One can then set down coordinates that freely-fall with you and your always-reading-no-acceleration accelerometers. In those coordinates, one could say that the rest of the universe is in motion about the coordinate origin, which is you. However, one would tend to reject the notion for reasons similar to the rejection of geocentrism.

It is however impossible to hold still in our spacetime. Our universe has a strong time-oriented causality and for good reason (including the behaviour of subatomic particles in countless laboratory experiments and astrophysical observations) we represent it as Lorentzian spacetime with certain constraints and energy conditions, and while that remains the best most fundamental representation of our universe it is safe to say that everything physical must be in constant motion through spacetime. So our freely-falling self-centred astronaut is only always at the 3-dimensional spatial origin of a set of 4-dimensional coordinates, one dimension of which is timelike. Indeed we can even say that minimizing the movement against spacelike axes, one must maximize the movement against the corresponding timelike axis. We are of course free to set down any set of coordinates we want -- doing so does not change the physical arrangements of matter, only how one represents those arrangements.