  We present an analysis of 1524 spectra of Vega spanning 10 years, in which we
search for periodic radial velocity variations. A signal with a periodicity of
0.676 days and a semi-amplitude of ~10 m/s is consistent with the rotation
period measured over much shorter time spans by previous spectroscopic and
spectropolarimetric studies, confirming the presence of surface features on
this A0 star. The timescale of evolution of these features can provide insight
into the mechanism that sustains the weak magnetic fields in normal A type
stars. Modeling the radial velocities with a Gaussian process using a
quasi-periodic kernel suggests that the characteristic spot evolution timescale
is ~180 days, though we cannot exclude the possibility that it is much longer.
Such long timescales may indicate the presence of failed fossil magnetic fields
on Vega. TESS data reveal Vega's photometric rotational modulation for the
first time, with a total amplitude of only 10 ppm, and a comparison of the
spectroscopic and photometric amplitudes suggest the surface features may be
dominated by bright plages rather than dark spots. For the shortest orbital
periods, transit and radial velocity injection recovery tests exclude the
presence of transiting planets larger than 2 Earth radii and most
non-transiting giant planets. At long periods, we combine our radial velocities
with direct imaging from the literature to produce detection limits for Vegan
planets and brown dwarfs out to distances of 15 au. Finally, we detect a
candidate radial velocity signal with a period of 2.43 days and a
semi-amplitude of 6 m/s. If caused by an orbiting companion, its minimum mass
would be ~20 Earth masses; because of Vega's pole-on orientation, this would
correspond to a Jovian planet if the orbit is aligned with the stellar spin. We
discuss the prospects for confirmation of this candidate planet.
