I was going to add this as a comment to dwjohnston's answer which I completely agree with. So first off: yes, absolutely do this. What I want to add is what it looks like from the other side. In my case, from the perspective of a software developer that was involved with a small-ish US contracting company and was heavily involved with hiring.
To begin, in my opinion, the largest costs of an intern is in the form of the time taken away from other employees training/coaching/mentoring them (at least in the field of software development). So even an unpaid intern isn't "free". We paid all our interns, but even free I suspect interns are a net loss in the short-term typically. The good news here for you is that a paid internship is quite likely if you get an internship at all. The bad news is that it can be difficult for a company to take on interns even if that intern seems like a great match. In my case, since we were usually subcontracted ourselves, in many cases it wasn't possible for interns to be billable meaning they would need to work on internal projects or projects that didn't essentially bill through. That means, while every hour I worked paid for itself, an hour worked by an intern did not which leads to cash flow issues. This is just another way of saying that hiring an intern can be difficult regardless of the quality of the intern.
The typical response for anyone who doesn't get past a resume screen is a polite, generic rejection or no response at all. This is what you should expect from most places you apply to. For the reasons I listed above and others, the answer will often be "no", and this is no reflection on you. However, if you applied to the company I was a part of, since we didn't have an HR department, it would be handled by someone like me, and I'd say the following more or less describes how any of the people filtering resumes would likely respond.
If your application didn't come across as a completely generic application that you were just spraying everywhere, you'd very likely get a personalized response from me. It would likely be a "no" for the reasons I mentioned above, but I'd certainly give you advice on when would be a better time to apply. I'd inform my coworkers about you or annotate you in our recruitment tracking system so that your resume wouldn't be filtered out if you applied later. I'd also give you advice on topics to self-study or focus on both for the career field generally and in ways that would make you a better fit if you did apply later. To be frank, initiative is so rare, that it is a huge plus to demonstrate it.
