I volunteered supporting people living on the streets.
The problem with using the -ing form in this particular sentence is that it's ambiguous.
It could be interpreted in either of these ways:
I volunteered to [support people living on the street].
I volunteered to [clean up the room].
I volunteered [supporting people living on the street]
to do something.
I volunteered [cleaning staff]
to clean up the room.
What a supporting person in the second interpretation would be, I don't know—and that makes it even more confusing than it would be if a more common noun (like cleaning staff) were used. But you'd be rounding them up and presenting them somewhere as volunteers.
The first interpretation uses the verb intransitively, while the second uses it transitively.
The exact sentence used in the question also adds to the ambiguity:
- I volunteered cleaning staff.
- I volunteered supporting people living on the street.
Although it's conceivable in the first sentence that you could have volunteered to wash staff members, it's unlikely, and common sense says that cleaning staff is a compound noun and volunteered is being used transitively.
But the meaning of the second sentence is far less obvious, so common sense can't apply as easily, and it remains more ambiguous than the second sentence.
However, context could make it more clear:
"I volunteered firefighters. Who did you volunteer?"
"I volunteered supporting people living on the street.
While this now makes the meaning clear, it doesn't prevent supporting people itself from sounding unnatural. I assume that it would normally be phrased differently:
"I volunteered support staff living on the street."
"I volunteered cleaning staff living on the street."
"I volunteered secretaries living on the street."
So, it's not necessarily the case that you can't follow volunteer with a word that ends in -ing. In some contexts, it's quite possible. You just need to be careful of those contexts where it's ambiguous or otherwise unnatural.