You'll definitely be able to get a folding wheelchair from a rental company, but I agree with Bete that you may have problems with size. When I had one as a loaner while waiting for my manual wheelchair to be made (I have a custom, rigid chair with a back that folds down), I know I couldn't fit it into the trunk of a Saturn Ion. My loaner was possibly smaller than a rental you'd get from a company (it'd been someone's custom personal chair that when they replaced, they sold back to their DME so he could sell it to someone who couldn't afford their own, but needed a similar set-up). I ended up putting it in the back seat, but it took up most of the floor space so you definitely couldn't fit 2 people in the back and probably would have been squished with even one. If the car wasn't 4-door, I wouldn't be able to have fit it at all. People get confused when I say this, but my rigid manual chair with the back that folds down flat onto the seat actually fits more places and often with a smaller footprint than that normally folding wheelchair did.
I have seen folding manual chairs fit in the trunk of a typical NYC cab, but those tend to have huge trunks. So if you got to the level of car such as a Ford Crown Victoria, it might fit. It should fit in most or all hatchbacks, as well, but I don't generally see those as a available in economy level by rental companies.
In Europe you get folding chairs, sometimes even rental ones, where half the back folds down, but I've never seen one in the US. It'd be nice if they had them like that because then they'd fit in many more cars!
