Should I remove the foam when cooking chickpeas?
According to a quick search it appears that your notion about it being protein-based is correct.  Most of the recipes I've seen say to skim it; the above-linked site says that adding a little oil will keep the foam down.

I personally wouldn't do anything with it as an ingredient unless I had a truly massive amount of it to experiment with--I don't know enough about the properties to make anything other than wild guesses about how it would work.  The only similar material I can think of would be beaten egg whites, but unless you were desperate for a vegan alternative and were already boiling up vats full of chickpeas I'd just use the eggwhites.