What does the evidence say (if there is any) on a) the safety and b) the efficacy of COVID vaccines in people with IgA deficiency? I assume both a and b may differ depending on whether it's an mRNA or old technology vaccine.

I don't know whether there are larger studies with immunocompromised patient cohorts, but small-scale studies show that the vaccines were well tolerated, but immunogenicity varied quite a bit. While some of the patients might have selective IgA deficiency, I don't know whether there are studies focusing on this conditions specifically. Generally speaking though, they would expected to have similar or better outcomes than compared to other immunocompromised patients.

See e.g. Ponsford, M.J., Evans, K., Carne, E.M. et al. COVID-19 Vaccine Uptake and Efficacy in a National Immunodeficiency Cohort. J Clin Immunol 42, 728–731 (2022). https://doi.org/10.1007/s10875-022-01223-7

Some other studies seem to show some level of protection, but it looks a bit different than in non-immunocompromised individuals and breakthrough infections (pre-Omicron) were a bit more common.