Submit Runs to speedrun.com

This used to be blocked by livesplit-core, but due to all the async changes happening in Rust, I'd like to implement the Web Communication with normal TypeScript first and later move that stuff over into Rust, once the Rust ecosystem is ready.

This issue specifically may however still be somewhat blocked by livesplit-core, as it probably requires changes in the Run Editor's state.

Start by reviewing the existing TypeScript web communication and the livesplit-core integration, then inspect how the Run Editor's state is represented. The issue does not name files or tests, and it does not define a completion condition beyond submitting runs to speedrun.com; clarify the required state changes and Rust migration boundary first.

Written by the indexing model from the issue text.