What's the process for a NG soldier to volunteer for a deployment with a different unit?

If you are Nationsl Guard/Reserve look up Tour of Duty on hrc's web site. You will have to use your CAC. Find a slot and then gain approval from your chain of command...then volunteer.

First off, unless your CO agrees, you are going nowhere. So do this: go to your PSG and/or 1SG and tell them you want to volunteer. If they can spare you from the TO&E they'll probably grease the skids for you with the CO. But if you are: a-mission critical, b-in a shortage MOS, or c-up to your eyeballs in alligators because the COC has it in for you, you might be stuffed. Unless there's a command initiative looking for volunteers, you have no one to appeal to above your CO. He's the one who must ensure your unit is ready to deploy -- whether it is scheduled to or not.

Deployments are either posted or word of mouth....posting is normally command approved...if you see the post, someone approved it....word of mouth...maybe?....contact the deploying unit first....they will handle it and or advise....