You don't get a separate paycheck from him, there is no paychecks actually issued any more in the military. For Navy the first pay can take up to 6 weeks to be paid, it will though be backdated to the date he started. He wouldn't have been paid yet, the earliest pay he would see would be on October 15th if he started before mid September. If he started after mid September then it wouldn't show up until Nov 1st.
If he put in the proper address & information then you should receive a letter from NFCU in the mail. They will issue him a Debit card in his name only and a separate PIN will appear in the mail as well. If he put his Boot Camp address then he will receive it. The only way you will have access to this money is if you get the Debit Card and PIN in the mail, if you don't then you are going to have to wait until he graduates. This is something you honestly should have been prepared for by the recruiter. If he didn't have a bank account before he left then he would be required to use NFCU or another one. If he did have a bank account and took the information with him, then he could use that account for pay.
Have you received an envelope in the mail yet that has a Form DD1172 for you to get a Military ID card? That comes around week 3 or 4.