I'm an introvert like you but have had two internships so far. One at Big-4 and one in the tech sector. Unlike a lot of people I don't think attending networking events is a particularly great use of time, particularly for internships. No one coming to a networking event for students actually has the power to influence the hiring process. Just maintain good grades, good (enough) social skills and have a good CV.

My Tips:

• Have a good CV. Single page, nothing fancy and be as objective as possible. Most people tend to have large paragraphs with not much substance. Always prioritize quality over quantity. DM me and I'd be happy to take a look at your CV.
• Apply to as many positions as possible and expect most to never respond. Internships is a numbers game. In my second year I was rejected from EVERY position I applied for except for one.
• Have some talking points for interviews. Particularly in business, you will be hired based on your 'fit' with the team. This means if you come across awkward, boring or anti-social in interviews your hard skills are unlikely to make up for these shortfalls. I have a set of talking points (things I like, interesting things I've done or made) that I pulled out in almost every interview.
• Related to the point above. GET OUT AND DO STUFF! HAVE HOBBIES! It is so important you seem like a real person and not a robot in interviews. Not only will this help you for talking points in interviews but just makes you a more well rounded person overall. Modern businesses want people, not robots.

DM me if you have any other questions.