I know a couple of people who have gone to sea with cutters. I've been invited, multiple times, by the 210 that I support to go to sea with them. Unfortunately, its never been doable for me, given work and family commitments.
The key issue is you have to qualify. If you want to qualify in an active duty job, you have to meet all the requirements. I know a guy in my AUX flotilla, who was a radioman when he was an enlisted CG member 35 years ago. He's retired now, and went and qualified as a communications watchstander. He now stands communications watch in the Sector radio room. Took him a while to complete the qualification. But, if you can meet the qualification standard, and do the job professionally - there are all kinds of things you can do as an AUX.
But yes - can and does happen that AUXies go to sea. It all depends on what the unit needs. If they have a spare rack, and there is something you can do that adds value, you can end up with an opportunity to go to sea.
If you live near the homeport for a cutter or buoy tender, you start by figuring out a way to volunteer for the ship when they are inport. Do that, so they can look you over and figure out that you will carry your weight and fill a need for them. Once you've worked for them a while, then tell the CO/XO that if there were an opportunity to go to sea with them, you'd be up for that.
On the AUX side - look at qualifying in the AUX-FS program. The Coast Guard service wide is worried about shortages of FS qualified members. Big issue for cutters. Do the AUX-FS qualification, volunteer inport for a cutter or buoy tender near you when in port, then offer to go to sea with them. AUXFS regularly gets calls from CG units needing help. Including requests from cutters/buoy tenders.