Can someone who is in the process of getting a green card (already submitted) join the military?

Needs green card

Wait how old is this guy?

Ok he can join the Reserves only with a Green Card, if he is naturalized then he can join active or reserves. So my question to you is: Why are you assuming the recruiter is lazy? You're only hearing one side of the story. Why don't you call his recruiter and ask these same questions you're asking the forum? Or better yet where is this applicant so I can ask him some questions "peel back the onion" and the true objection.

AR 601-210
(2) For aliens. Applicants must present their INS Form I–551 (Permanent Residence Card) showing that the person
has been admitted to the United States for permanent residence. INS I–551 cards issued after 1989 are only valid for 10
years and must be renewed. Additionally, any INS I–551 card with an expiration date within 6 months of the accession
date must be renewed. Applicants with expired cards keep their permanent residence status; however, they must apply
for renewal of their permanent residence status INS I–551 card and must obtain verification in the form of an original
receipt from the U.S. Citizenship and Immigration Services indicating that the applicant has paid for an INS I–90
(Application to Replace Permanent Resident) renewal application prior to shipment to training. INS I–151 cards are no
longer valid and applicant must obtain an INS I–551 card. Resident alien cards without the picture of the applicant are
invalid for enlistment purposes. Resident aliens must provide proof of place of birth.

SPC(P) (Join to see) does he have his heart set on the Army? There are several jobs for Officers in the Navy that will accept him with his age and naturalized status.