Each college has a drama society. The university drama society is called OUDS. You can pitch for funding to any of the societies. Sometimes two or more college drama societies co-fund a production.
You advertise in Daily Information and/or on social media for actors, tech crew etc, and conduct auditions. You negotiate with your own college or another college to use a space in the college (indoor or outdoor), or you rent the Burton-Taylor Room* or another venue. You can rent equipment, costumes, props etc from Bermans, the RSC etc, or make your own.
Most plays run for a week during term time, and maybe longer if aiming at the tourist market with a summer garden production.
There are a bazillion clubs and societies at Oxford covering all known and some unknown sports and pastimes. Many are reasonably well funded. Get involved!
*Richard Burton, one of the greatest actors of his generation, loved his time at Oxford, and he and, after his death, Hollywood Goddess Elizabeth Taylor, who was still in love with her twice-ex-husband Burton, endowed a performance space called the Burton Room, later the Burton-Taylor Room.