Can your unit/leadership prevent you from volunteering for SF?

Paperwork is often lost, intentional or otherwise.

If you are having trouble with this, talk to your SF recruiter. A conversation by your recruiter with your leadership could help, since he isn't in your chain of command. This may have changed, but there wasn't any paperwork that the unit had to see or sign for a Soldier to volunteer for SFAS, so nothing for them to lose, misplace, or fail to sign. Good luck, and remember that everything you do to get to an ODA is part of the test, including getting your packet approved for SFAS.

Short answer: No!

Long answer: No, but can delay you due to operational requirements.

If you are serious about attending SFAS, perhaps a nice chat with a SF a recruiter is in place.