The whole idea of modern version control is that every developer (every user) gets his/hers own copy of project, his/hers own working copy (working area, worktree). This way one developer work is separated from other developers, and won't mess others work. (Of course version control system has also to have an ability to merge, i.e. join changes).
The whole idea of distributed version control is that every developer (every user) gets his/hers own copy of repository, his/hers own clone. This way one developer can work on series of steps, series of commits separated from other developers.
Note that if you have already cloned repository on your computer, all Git GUIs allows you to select existing Git repository (sometimes by launching from inside said repository) instead of creating fresh clone. Of course then you should remember to keep yourself up to date via fetch / pull, and keep your remote publishing repository up to date via push.
That said some git hosting sites / software forges include in their web interface ability to manage / edit repository from web browser. For example GitHub provides interface for creating, editing and other operations on files, creating branches and merging them; you can even provide change proposal on foreign repository via GitHub cloning and creating pull request for you.
Beanstalk also (from what they say in documentation) has some support for code editing (at the bottom of page, left column).
There are also tools like GitHub for Mac, but I don't know if it uses GitHub API to edit remote repository like via web browser, or like other Git GUIs require you to clone repository to desktop.
Of course it wouldn't work that way with Beanstalk-hosted repository; Beanstalk has thought some third party tools like Habichuelas or Magic Bean for iOS.