How to initiate an application of a research attachment fellowship directly with the potential advisor?
If they expect you to get in touch with faculty first, and they widely accept students from beyond CalTech, then they're expecting you to reach out - no undergraduate has a particularly robust academic network beyond their institution (beyond a few edge cases).

Generally speaking, the best way to do this, in my view, is to email the professor, give them a brief introduction about yourself, your research interests, that you're applying for the SURF program, and that you'd like to work in their lab. Link this to the work they do.

I'd consider doing this for several professors, unless there's just one whose work matches your own. Worst case, they'll ignore it, but professors often at least read these, especially if they're from institutions they recognize, have concrete information, and are in coherent English.

For the record, this is how I got my first undergraduate research position, and ended up with a wonderful colleague I work with to this day.