Where exactly at Hampton Court Palace can you get a cab to Kew Gardens?
Do you have to call for a cab or do cabs drive buy you can take?
What cab company do I use?
That would be a very expensive cab ride.
Why aren’t you taking a bus?
Personally I’d take bus 111 from the Chesnut stop
A outside of Hampton Court Palace to Richmond.
Kew Gardens is a 15 minute walk from the Richmond station.
One more option:
Take Thames River Boats from Hampton Court to Kew.
A most scenic way to go. There's a pier at Hampton Court and one at Kew. Easy.
https://www.thamesriverboats.co.uk/thames-boats/kew.asp
Review the information on their website for the boat schedule.
However, if you still want a taxi, ask at Hampton Court Palace. They may have a recommendation. I would expect you'd have to contact the taxi. Unlikely that you'll see one driving by.