We study the phase behavior of multicomponent lipid bilayer vesicles that can exhibit intriguing morphological patterns and lateral phase separation. We use a modified Landau-Ginzburg model capable of describing spatially uniform phases, microemulsions, and modulated phases on a spherical surface. We calculate its phase diagram for multiple vesicle sizes using analytical and numerical techniques as well as Monte Carlo simulations. Consistent with previous studies on planar systems, we find that thermal fluctuations move phase boundaries, stabilizing phases of higher disorder. We also show that the phase diagram is sensitive to the size of the system at small vesicle radii. Such finite size effects are likely relevant in experiments on small, unilamellar vesicles and should be considered in their comparison to theoretical and simulation results.