Do I have to wait to reenlist before submitting a Recruit the Recruiter packet?

You NEVER have to reenlist prior to be accepted to any school or submitting any packet. You simply have to be eligible to reenlist when it's submitted and once you're approved you are required to reenlist within 30 days. You can't go to ARC and then reenlist.
Also, if you're eligible for a bonus you can reenlist outside your window when you come on assignment.
You can find this in AR 601-280 under Service Remaining Requirements (SRR).

You do not that I am aware of. You will be expected to reenlist or extend to meet the 36 month obligation. It might be before you go to school or while you are in school. I have seen both. Granted if you are going to volunteer you might as well reenlist or extend before submitting if your window is open.

AR 601-210 is your source I remember people having to reenlist in school to meet requirements. That was a while back I would tell you to look at the USAREC website for FAQs

Go ahead and submit it, you won’t be denied, you’ll just have to re-enlist so you can get scheduled for a school date.

I’m not 100% sure, but they may approve your packet and make it conditional on you extending or re-enlisting.

Submit your packet now if you want to. When you’re on orders you’ll be expected to re-enlist or to submit a dec-statement. I was in the same situation when I was voluntold to be a recruiter.

Normally you have to either re-enlist if you’re in your window, or extend to meet the 3 years...

Good responses and yes AR601-210 is the regulation that I helped write when I was at DA. Hooah and Good Luck. I too was a former Recruiter and USAREC ops officer.