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

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.

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.

If you know of a unit that is heading out the door and has an open slot for your rank and MOS, then you need to speak with your Command to see if you can get transferred to said unit for deployment. Bear in mind, your unit is not required to release you.