There are numerous accessibility problems with the file picker.
1. It is not clear to screen reader users that a modal window has popped up when the file picker starts.
2. The focus is initially put on the last item in the file picker so the user must reverse tab through the window to access the file picker functionality.
3. There is no semantic structure to the file picker to denote that the left side of the window is where you select your repositories and the right side shows you what is in those repositories or lets you execute other actions on the repository. The only logic to its layout is through visual cues.
4. It is not apparent to a screen reader user that when a repository is selected on the left hand side of the window that something actually changes on the right side of the page.
5. It is easy to accidentally tab to an area outside of the file picker modal window so that the user is no longer interacting with the file picker.

To solve these problems there are a number of patches that could be applied, or the file picker could be re-engineered completely.