I agree with kcoughli, contact the programs.
I know for me, I graduated early May and despite submitting my state application 2 days later (literally the day the paperwork was finalized through the school), it still took almost 6 weeks for the license to be approved. Mine was approved two days before my internship started. A couple of my internmates didn't get their licenses approved until the end of our first week, and it was fine, they just couldn't sign off on anything on their own. With that said, June 15 is a pretty common start date for internships so if you can't apply for your license until after that date plus the processing time with the state boards, it could be an issue for some programs. But maybe other programs don't leave interns alone immediately so they can work around it. Just contact them and ask.