We present an analysis of 1524 spectra of Vega spanning 10yr, in which we search for periodic radial-velocity variations. A signal with a periodicity of 0.676day and a semi-amplitude of ~10m/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 activity signal appears to evolve on long timescales, which 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 10ppm. A comparison of the spectroscopic and photometric amplitudes suggests that 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 2R{Earth} 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 15au. Finally, we detect a candidate radial-velocity signal with a period of 2.43days and a semi-amplitude of 6m/s. If caused by an orbiting companion, its minimum mass would be ~20M{Earth}; 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.