I'm going to reject the premise of your question outright. Windows should be flashed properly and the siding should have a drain plane installed behind it. This isn't a job for caulk.
Some thoughts that may be helpful:
- In modern construction, sill flashingsill flashing and house wrap provide the actual weatherproofing around and below windows. Your home appears to have neither.
- Vinyl siding provides visual appeal and protection against physical damage. It does not offer much in the way of weatherproofing. Air and wind-driven rain move freely around it.
- That gap is too large for caulk, and the vinyl channel is too flexible as a base for caulk. It would look terrible, and it would likely fail.
- Vinyl siding is meant to move as it expands and contracts due to temperature, and rarely should it be caulked.
I'm afraid I don't have a great solution for you here. Your siding replacement was done very poorly, and there are no shortcuts to a resolution. Your siding contractor failed miserably, in my opinion, by not providing a path to a viable outcome.