Great question. It is currently not officially supported. When you start an Experiment the web UI sends a request to our swagger API of the orchest-api container. If you look at that request you can see how you could trigger a run programmatically.
It's planned for the future, but we haven't yet documented or designed a stable API for it. I'll look it up soon to post something you can use here. Want to doubly emphasize it's not a permanent solution ^^