We were also told the same with his egg allergy as most vaccines are cultured in egg
We skipped any with allergens because our family medical history made the risk too great for something that generally only causes a rash in otherwise healthy children, but you need to do what you're comfortable with. My oldest has been through two outbreaks of measles in our State with no issues. You may be required to do them eventually for school anyway, if your state/country doesn't have an opt-out option. I haven't heard of any alternatives to a vaccine, other than immune serum globulin (if you're already infected) and keeping your immune system strong during an outbreak. Some take Vitamins A, D3/K2, C, zinc, etc. year-round.
I questioned this when my eldest had his vaccines and the nurse said it was a different protein to what they drink if that makes sense but to be sure they made us wait in the waiting room for 30 mins once he had the vaccine just to be sure