You probably could. I personally wouldn't attempt it because I mostly do software and wouldn't want to risk draining it dry overnight or voiding the warranty or something, but if it's a just a car battery it shouldn't be hard for someone with more EE knowledge.
Another option might just be to mount an entirely sperate power system just for the phone and any other devices. Buy a huge USB power bank off Amazon and mount it to the carriage or something, might not even need duct tape.
That was my first thought too. If there's a 12v battery on the chair, a car USB adapter could power the qi pad. Then the issue is how to move phone to pad and move it off the pad for use. There would be virtually no chance of running down the wheelchair battery with this setup; the motors use much more energy than a USB charger.
Another option might just be to mount an entirely sperate power system just for the phone and any other devices. Buy a huge USB power bank off Amazon and mount it to the carriage or something, might not even need duct tape.