I was just promoted to E-5 as of 01 July. How long will I have to wait to submit a recruiting packet?

Look at the USAREC website: https://www.usarec.army.mil/hq/recruiter/volunteer_recruiters.shtml. However, I strongly recommend you to wait until you gain NCO experience in your current job before applying. You really need that strong foundation in your MOS at your current level. Remember that you will lose a couple of years of experience in your job if you get selected. Eventually, you will come out of that assignment (unless you convert to a career 79R) without a solid MOS background and you will probably be a SSG by that point.

In summary, think and plan ahead before pulling the trigger.

Depending on your MOS you may have to serve time in certain leadership positions before they’ll consider you for recruiting.

If you have reenlisted at least once, contact your branch manager. He/she will give you further instructions.

Not sure. But as a new Leader, try and get leadership time in to grow and get experience before recruiting. That's my opinion.

Contact the G1 at division to speak with the USAREC “Recruit the Recruiter” rep. They will give you a template. Alternatively, you can send the volunteer packet to USAREC directly but don’t be surprised when you hear zero feedback until it is accepted/denied.