My son is 171/2 and he wants to move out of my house would he then get the support payments I get from his mother?

Support would end if he lives on his own. He would be considered emancipated. The payments would stop.

The child does not receive any money, the parent raising the child does.
So no he will not get the money.

First I doubt at 17 1/2 you can even let him just move out on his own, not at least without the other parents approval also. But if allowed, in almost all cases any child support ends.