A lot of the ambulances on phuket are owned and managed by charities or foundations, besides collecting injured people after road accidents they are used to move patients around from hospital to their house.
They should be able to meet your requirements, I don’t know if there is published price or if you make a donation to the charity, you will have to discuss it with them.
The large charity I know working in phuket is the Kusoldharm Foundation and their number in phuket town is 076-246301.