It's a rest API, so you might be able to some of it with the AJAX plugin and use of the JSON object.
If that doesn't work, you can use JS C3 scripting to access the API (using fetch / XMLHttpRequest, etc.). It seems definitely possible. I looked at the SR API github, it has not been updated in a few years. It might be good to make sure it's still working and operational before doing too much work.
It does look a little clunky, you may need the user to copy and paste their special auth keys and save into local storage (but you can perhaps do this 'in game' with an iframe, etc.)
Here's a library that you may be able to use for this (JS):
github.com/SwitchbladeBot/node-speedrun (would need to package for web user browserify or something like that.) Also note this library only 'reads' values from the site. It does not have functions to post runs. The full Speedrun API supports posting runs.