Do you have a professional, early-career website? I hope so. But if not don’t worry, just make sure you have all your portfolio materials ready at hand: writing and/or multi-media and/or video work samples; up-to-date CV; and please, do know how to write a cover letter. (If you need help with any of these things, I suggest you make an appointment at the campus career centre, known as “Career Boost.” The people who work there are trained professionals dedicated to helping and supporting with best current practices around CVs, cover letters, work samples, and the lot. Please do avail yourself of this excellent on-campus resource. One of the brilliant things about our uni is that we have the most qualified and up-to-date career professionals on hand to help you in a million different ways.

Here’s a link to the Career Centre: )
Alright, you’re ready to make your Approach. Here goes!
Start with an email, and use a subject line along the lines of  “Seeking Internship.”
In the body of the email, introduce yourself, and say that you’re hoping to intern at (the particular place) because … (and do ask yourself the question Why? You should already have a reason, e.g., you like their programming? you have watched/read/listened to them for years! Or, in the case where your Approach is to an organisation you don’t really know, don’t try to fake it. Just say it looks SO interesting! And you could be a good fit! And may I come for an interview!? You know what I’m saying. Just let them know what is prompting you to make this approach to this particular placement, and that you’re excited about the prospect of working with them.
And avoid saying “I HAVE to do an internship for my 4th year … etc.”  It makes you sound a bit desperate. Rather, tell them you are looking forward to your internship next semester. Do be absolutely clear at this early stage about whether you are seeking six weeks full time OR 12 weeks part time. Provide the dates you are available! (please ask me if you’re not sure of those dates).
You should mention it’s for academic credit. Include a few words about what you’re interested in, and good at, where your strengths lie; where you get your news, or how and where you are engaged. And … would they have a spot for an intern during that six or 12-week period? If so, would they now like you to send them a proper covering letter, CV, and perhaps some work samples? Would they like to meet you for an interview?
And take it from there.
IN GENERAL … If you have made an approach and they have indicated they are interested, then please allow them the time to take you through their normal process, and to make a formal offer (or not) BEFORE you go off seeking something else. On the other hand, they will (or ought to) understand that you need to find something, and can’t afford to sit around and wait for them. They owe you the professional respect of a timely response. And don’t forget, at this stage of your university career, it is all about professionalism, yours and theirs.
Of course the thing you must NOT do is accept a placement, and then say … oh hark! … another one has come along and you like it better, so you give up the first one. Don’t do that. Just don’t. I will look very unkindly on that kind of thing. Very unprofessional.