Could he get multiple free attacks in a single turn?

Or is it one of those powers that does a single check to see if there were ANY 4 damage hits at all, if yes then it triggers just the one time.

It doesn't mention "If another friendly character with the Avengers keyword and a damage value of 4 or more hit an opposing character this turn.." Which is often their way of wording a power that does a single check during a turn.

I guess the wording of "hits" makes me think it could be each time friendly character hits.

It tries to trigger every time another friendly character with the Avengers keyword and a damage value of 4 or more hits an opposing character, but because you can not activate the same effect with a free action more than once in a turn, he can only get one free hit per turn in.

2014 rulebook page 9:

A character
may not activate the same game effect
twice in one turn with free actions, unless
the game effect indicates otherwise.

You could decide not to make the attack after the first such qualifying hit, though, saving it for (hopefully) another hit later in the round.

You are not required to take the Free Action immediately after the triggering attack is completed. He has the option of spending that Free Action for the rest of your turn.

Oh geeze. It's just sitting there and I read right past it.