Sponsoring a non-academic collaborator to attend a conference
Is there any provision for my collaborator's university to pay for my travel costs as well?


While not strictly impossible, this sounds rather unlikely to happen, at least through formal channels. Travel budget is restricted, and there are often even administrative restrictions that prevent faculty to give it to people who are neither employees nor students of the university.

That being said, if your collaborator is sufficiently commited to making you go, as well as sufficiently crafty to bend the rules in the right way, (s)he may cook up an arrangement with the administration of some sort. Note that this will likely be (at best) semi-officially, and will burn some real or political capital of the faculty, so (s)he won't just do it to be nice. There has to be some real benefit to them that you are at this conference.