No, there is not upload function for Runs or Swims at this time. There is an open Feature Request (I can link later when I have more time), and the TR crew acknowledged this is on their overall to-do list, but not active at this time (because other priorities are ahead of that implementation).
Manual is the main way to get them in the TR system right now.
I had a cheat workaround I used a bit once upon a time. I will try to find that link too.
Edit: Feature Request:
Related Post:
My odd work around, that is not really a good idea, probably: