My son got immunization shots in each arm and now his whole arm is swollen.is that normal?

See below: Can be, depending on the immunization. I occasionally see a lot of arm swelling, after kids receive their 4th and 5th dtap.

Yes: Yes, it happens.Occasionally the shot can cause local irritation. I find that heat applied helps, altho some suggest the opposite, ice.Either way, while annoying, not serious.This should not prevent immunizations.