In fact both sentences are incorrect, because the correct tense to use is "allows". "As much as" is also wrong. You can use "as long as" or "so long as" These are correct: I'll do it as long as my health allows. I'll do it so long as my health allows.

The mean slightly different things, and to me both are incorrect. "I'll do as much as my health allows (me to do)." "I'll do it as long as my health allows (me to do it)."

I would say yes but the second sentence definitely sounds much better.