Im a part time taxi driver.
Yes it’s normal, needs to be pre-arranged and agreed with the school and Taxi firm as you need a regular allocation of driver for the kids to be released to them. This should be in writing to both, I’d suggest the child should also keep a copy in bag incase of substitute teachers. Speak to the school first as they may have a council contract already that may make it cheaper for you (if the vehicle is in the area and has space), or you could use the same firm.
All taxi drivers must conform to the local and District councils licensing, so will be DBS / CRB checked, In Gloucestershire we have to go on a safeguarding course in addition which specifically deals with kids and vulnerable people.
If it’s a good firm they’d send around the driver/s at a convenient time to meet the kid/s and discuss the arrangement with you. If they wont do this, don’t use them. You’ll need to take their personal taxi/private hire licence number to put in your letter as well as the operators licence number and telephone numbers. If you or your child is not happy with the driver, for any reason, ask for a different driver. You could ask for the drivers mobile number, they may or may not be happy to give this to you.
I’d also suggest a fixed daily rate payment via direct debit or weekly/monthly invoicing rather than the meter or giving the child cash – safer for driver as well as child.