I can relate a lot to what you've described; I've always felt uncomfortable in large social gatherings, especially with people I'm not familiar with. When interning at the company I work full-time at now, I knew of several interns who acted like you described (i.e. getting lunch with random coworkers from different departments), and I never felt comfortable to do the same. In my experience as an intern, it matters more to make connections with your mentors/teammates who you're working with over the course of your internship, since they're the ones who'll be providing the majority of the meaningful feedback that determines whether you get a return offer. That being said, it can be worth going out of your comfort zone sometimes! Meeting new people is something that will happen a lot over the course of your career, so my advice would be to focus on making a good impression when you do happen to meet new people rather than going out of your way to make a bunch of connections. Barring a toxic work environment, you'll do fine as long as you have a good relationship with your teammates and immediate superior and don't actively alienate anyone else.

(For what it's worth, I've seen quite a few threads like this that seem to get responses saying that the OP has such and such psychological condition; it kind of astounds me how so many people a feel that it's prudent to provide a stranger with unsolicited medical diagnoses based on a couple paragraphs they posted on the internet. You're under no obligation to respond to them, so feel free to disregard them if you'd prefer to.)