You can. Orange County and many of its communities offer free courts to residents and nonresidents alike. (So do other counties, but that's for another time.) If a court is listed below, it's free unless noted. The county and cities assured me that to play, you do not need to show an ID. Let me know at email@example.com if that turns out to be incorrect.
All free courts in Orange are mapped at this site.
Orange CountyFree-courts map here:
The Orange parks with tennis are free except Fort Gatlin Recreation Complex and Shadow Bay Park. They are:
Barnett Park, 4801 W. Colonial Drive, Orlando, FL 32808
Blanchard Park, 2451 N. Dean Road, Orlando, FL 32817
Fort Christmas Historical Park, 1300 Fort Christmas Road, Christmas, FL 32709
Orlo Vista Park, 1 N. Powers Drive, Orlando, FL 32835
Turnbull Park, 4941 Turnbull Drive, Orlando, FL 32812
Warren Park, 3406 Warren Park Road, Orlando, FL 32812
Chapin Station (West Orange Trail), 501 Crown Point Cross Road, Winter Garden, FL 34787
Wheatley Park, 2 W. 18th St., Apopka, FL 32703
Goldenrod Park, 4863 N. Goldenrod Road, Goldenrod, FL 32792
Info on Fort Gatlin's rates: fortgatlintennis.com
Shadow Bay Park courts are $6 per hour.