The built in seats are for forward facing children, not infants and the only vehicles I know for sure that they are available for are the mini-vans (Chevy's and Pontiacs for sure, not sure about the others) As far as the vehicles having them, they are standard on some models, but it would be hit or miss as to if the agency carried that model and if they do if they had one on the lot at the time you rent. The only car I can recall them being in was I think the Dodge Intrepid and that was a few years ago. I haven't seen any cars since with them in. I would not count on their being one in the vehicle. Also, even if there is your child might not like it. My DD had to ride in one for a few miles (she got sick all over her carseat) and she HATED it the whole time. She was 2 1/2 at the time and very vocal about her dislike of the seat. If it were me, I would bring my own, you know it's history, some of the rentals do not seem very nice at all.
Also, did they tell you why they won't let you buy a seat for a child under 5mo old? I never knew any airlines did that, I would think they would want to sell you a seat for your little one. Of course, I don't fly much, nor have I ever flown with a child that young.