ABSOLUTELY.
Especially vitamins that come from food sources (best as far as absorbability goes, but worst as far as allergic reactions go)... for example people who are allergic to shellfish can't take calcium coming from oystershell (the most common source), but can very easily take calcium that is gotten from a non shellfish source.
So, too, the binders and "inactive" ingredients can be allergy sources.
SLIGHTLY different, some people have disorders which make them unable to process certain vitamins/ minerals.. or do so badly. My uncle-in-law can't process vitamin A. Born with completely healthy eyes and organs, he's been blind and "crippled" because of vitamin A toxicity since he was about 4 years old. He has major kidney and liver problems, as well as a few other things that come from vitA overdose. My neighbor has Celiacs. She can't take most vitamins because there are trace amounts of wheat in them. Personally, I can take vitamins just FINE (all kinds) EXCEPT prenatals. The massive dose triggers migraines. Kids vitamins, however don't blip my radar at all, and actually help PREVENT my migraines. My mum is prone to kidney stones, she cannot, cannot, cannot take anything with calcium in it. Her body is flush with Ca++. She can't even eat spinach. Her bones and blood Ca++ are great... her body just *yanks* the calcium out of anything with Ca++ in it, and then ends up with an overabundance, which calcifies into stones in her kidneys.
So, yes. He could be allergic, or have a more serious problem, but since the rash is new from the vitamins, I would suspect it's merely an ingredient in the new vitamins.