"He is John's dad, whom may be coming." -- is this sentence grammatical?

It should be "who" because John's dad is performing the action of "coming". That is "John's dad" is the subject of the verb "coming". "Whom" is used for the object of the verb, for example "It was John's dad to whom I sent the invitation".

Having said that "whom" is not used much in ordinary conversation these days, most people would say "It was John's dad who I sent the invitation to" or, more likely, "It was John's dad that I sent the invitation to" or even "It was John's dad I sent the invitation to".

If in doubt use "who" in ordinary conversation. Using "whom" incorrectly would be a big mistake and even using "whom" correctly can make you sound pretentious.lk