She can have all her vaccines, including MMR and influenza. If she is going to sub-Saharan Africa and needs a yellow fever vaccine, BM me and we can talk.
You are very likely correct that she is egg allergic.
Your pediatrician can do an ImmunoCap for egg to find out (blood test); or she can see an allergist. No epipen needed if only hives. Keep liquid Zyrtec on hand and give her 5mL if she accidentally eats egg. Baked egg in cakes, etc is usually fine.
--allergist