Camping near San Francisco (without car)

Angel Island is just a ferry ride away. Check it out! http://www.reserveamerica.com/camping/angel-island-sp/r/campgroundDetails.do?contractCode=CA&parkId=120003

Check out Muir Woods, amazing camping grounds and great nature. 

They have season bus shuttles there.

http://www.marintransit.org/routes/66.html

511.org is the online transit planning site for SF. You input your location, where you want to go and the site provides alternate routes. 

I believe you can camp north of SF in Point Reyes Station, or south of SF in Pescadero. I know Pescadero better and have camped at Butano State Park and Memorial Park. Both are beautiful Redwood Forest parks. 

You will need to do the research to find out if they are served by public transit. Using Uber to reach these locations would be expensive, as they are somewhat remote. Also you need to book the campsite in advance via the state parks site, as camping is popular and sites book up in advance. 

I don't know if there are campsites closer to SF or even within the city. You are allowed to picnic at night and have bonfires at Ocean Beach, but I don't think camping is legal. 

Good luck with your trip.

Be prepared that we're in for a wet April.