I got rejected at a mid-sized tech company with ~2000 employees for a security engineering internship after the final round about 1 and a half months ago, but the recruiter said to keep in touch for future opportunities at the firm and also voluntarily gave me feedback in the interview process.
She gave me the biggest reason I was rejected was because there weren't any projects available that fit me at the time. I also have my hiring manager's email as well from the final round interview and he told me at the end of the interview to keep in touch.
Should I reach back out to the recruiter and/or hiring manager to check up on their projects and ask for a potential internship in the Fall or Winter? If so, what should I say in the email?
I'd reach out if:
Either of the above forms of communication will "move the conversation forward" instead of just a less-exciting question "hey, is there a job available for me now?"
Note that you can manufacture the above forms of urgency. For example, you can say that you plan to take a class in the fall which will close enrollment in 2 weeks, but you'd be more excited about an internship at Intel.
Finally, I'd also be cautious about fully trusting the recruiter's feedback. There's no reason she'd lie, but also there's no incentive to share everything. They may have zero intention/headcount to hire you in the fall/winter. See this discussion: Getting Feedback from Recruiter after Rejection
It's not entirely similar, but I recommend watching the lesson about cold messages in my job searching course as a lot of the same concepts apply: https://www.jointaro.com/course/ace-your-tech-interview-and-get-a-job-as-a-software-engineer/cold-messages/
Overall, you just want to show that you've put in the effort and aren't sending a cookie-cutter message begging for a job. As Rahul mentioned, if you can find a project/space that company is investing more in (as a 2,000 person company, they will post updates to LinkedIn/Twitter/etc), then reach out to the hiring manager using that as a basis. From there, show why you're qualified to work on that area (internship experience, relevant coursework, side project/open-source, etc).
You should also look into the hiring manager themselves. Is there something you remember about them that you can bring up? (e.g. "Are you still playing pickleball every weekend?") Maybe they posted something recently on LinkedIn (e.g. "Recently read your post about effective 1 on 1s, and it was super helpful!"). In other words, show that you care about them as an actual human being, not merely a vessel that hands out jobs.