Something that I have noticed on LinkedIn is that people add their internships for the summer before they start them. I have found an internship for the summer but am now looking for fall internships and I think that putting it on my resume will make it much stronger but am not sure if putting something I haven't done yet is allowed. What would be the best way to go about this.
If you get called to interview, you should expect to have to answer questions about all the great things you accomplished in this internship you haven't yet done.
Expressed more seriously: a résumé is quite literally and by definition a summary of things you've done. It is out of place to list things you have not done.
As an interviewer, I wouldn't particularly ding an entry-level applicant for this, but I'd suggest you make sure it's very clearly differentiated from the actual-experience information you present.
I think that putting it on my resume will make it much stronger but am not sure if putting something I haven't done yet is allowed. What would be the best way to go about this.
Generally speaking, posting on your resume something you have not done is deceptive, and something not suggested to do.
I would only add something to my resume when I actually finish it.
Imagine (hopefully not) that your internship ended before the expected time. Then, having included it on your resume before actually doing it would not be truthful.
I'm not sure that Linked In, specifically, has this functionality, but a way to denote this without being deceptive, but to be able to put the link out there earlier rather than later is to say something like:
Accepted for (blah blah) internship for Dates X through Y
And put it under "other awards/experience" rather than positions you've held. You did, after all, positively attain the offer. And some internships have spiffy program names you can include.
As a side thought ... I wonder what use it is to do internships back to back in this way...? Back in my day (boy, do I feel old!?) - if it was separate internships, there was some terms of school work in between, the idea being that you work a little, it gives you some useful experiences, you build on that with your studies, rinse and repeat.
If it's purely that you need you need to get X terms/months of internships accredited to you to graduate, then also think about a 6 month internship over a 3 month one - you'll get better, deeper experience from that, and something more meaningful to talk about in job applications. At lease in engineering, I know bunches of places that would also prefer 6 months, for the same reason - you get more value out of the time with the person.
I never had the issue you mentioned, as I did ~1 internship per year, so I could talk about my past internship when interviewing for the next one.