I used MK taxi (with Ramp) in Fukuoka. You may need to judge if the car is big enough when luggages are loaded.

Try MK taxi

I used to courier luggages between locations. Very efficient and reliable. However, I am not sure that that accept foldable wheelchairs. If yes, that would be great.

I think you have to do it over the counter. My friend told me that telephone reservation is possible but you may need to speak in Japanese.