It's a good idea to look for internships with small organizations. Big companies usually like to have everything on file and they normally ask for a SSN. I have done two internships, my first one only required to submit a resume and it was a non-paid internship. The current internship that I'm doing I got through my University and it's a paid one. Oddly enough they only asked for an interview, not even a resume.
I would probably avoid internships that are close to the government, or ones that deal with kids because they do background checks if you want to work with kids, it's kinda dumb, but they keep people like k-man away from our kids. lol jk
Yes, I typed all this just to set up a good joke. Good advice nonetheless.
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DACA Application sent 08/17/2012 to Arizona Lockbox via Express
Delivered 08/20/2012