This from the Recreation page in the "Saratoga Tipsheet", available at check-in:
There are two clay courts located near The Carriage House. Hours are 9 a.m. to 10 p.m. and they are open to all Resort Guests. Resort ID is required. The courts are complimentary and are on a first-come, first-served basis. Room card will be held for racquet rental. Equipment is available from Horsing Around Recreation Rentals.