Rummy online is available In a no download version as well. You can find several websites in which you can play rummy against other players in the same websites in a no download platform but only for points and for fun and not for real money.
The only website which operates a platform for the Gin Rummy game, and only Gin Rummy is
Red Hot Rummy which allows you to play Gin Rummy against other people from all around the world for Real money or for fun and also play rummy tournaments and freerolls – all that with no download required.
All you have to do is a quick sign up, username and password and you can start playing instantly.
Other internet rummy games are available at the local computer game retail stores.