Can you submit paperwork to go to Reserves from active duty?

You can transfer into the reserves after your ETS. You'll meet with a Reserve counselor six months from ETS to discuss options

I'll add that your Anniversary Year starts the day after you ETS. That's important as you'll need a minimum of 50 points in that year for it to be counted as a "good year" towards making your 20. If you dither and take months to affiliate and start doing drills, you can lose out and will have to do it an extra year on the backend to hit 20. So make sure the counselor goes over everything and you work to affiliate with a unit ASAP. Depending on where you want to live, picking up a reserve billet ranges from easy to hard. When I did my Skipper tours, I'd get a list of eligibles. Those who can show up get chosen first vs. ghost reservists. That and more opportunities to be seen by those who matter doing your evals.