Can active duty request to join a deploying unit?

Sorry, that's not how that works. Your best bet, if your unit is not slated to deploy, reenlist to go to a high optempo unit like the 82nd, 101st, 10th Mountain, or 3rd ID.

SPC (Join to see) if you aren’t in a unit scheduled to deploy, work with your NCO support channel to find a WIAS tasker through G3 channels. World-wide Individual augmentation system. These are individual personnel requirements “taxed” across the Army to fill Joint manning documents and high priority missions. The other option is to see if there is a unit in USAEUR unit short your MOS and grade through the G1.

The real question will be if your chain of command will give you up to another unit, with or with out a trade.

Make your desire known to your command. The Operations channel (G3) will be best postured to know who is deploying when. You can reach out to that unit and see if they are short your MOS and do a By Name Request. The answer may still be no.

There are a few resources. I am taking an educated guess as I wasn’t Army.

Go up your chain of Command, talk to the monitor for your MOS, and you can also coordinate via your units career planner.

Deploying units are always looking to too off their rosters going into a pre deployment training cycle.

But if you really want this be prepared to be required to extend your contract to allow for enough time for training, deployment and post deployment crap.