How does the process work to reclassify to 25B? What kind of paperwork does it entail?

When you're in your reenlistment window, if you meet the line scores and have a clearance, and if there are class seats, depending on the current strengths of each MOS at the time, you can reclass. No class seats available, no reclass.

Your Best Bet is to get with your retention NCO. They'll know the latest In/Out calls and the process to reclass.