Am I allowed to do the work for a research assistant (RA) without having been awarded an assistantship? I recently finished my undergrad degree and'm now enrolled in an entirely online graduate (M.Ed) program in educational technology/instructional technology. 

Nonetheless, I'll be moving to the physical campus of this Lower Midwest flagship university in late August. Although I'll be receiving grad Stafford loans (huzzah, no undergrad loans!), I applied for financial aid past the assistantship deadline.

I know the importance of networking and collaborating with faculty. I'm prepared to live off of loans (which is feasible, given the low cost of living in this city) so that I can work for one of the projects associated with my school. It's my hope that this would make me an attractive candidate for a 2016-2017 assistantship.

So...should I live off loans (and, realistically, a 15-hour/week part-time job) for that first year, so that I can be what's essentially an unpaid intern?
It's a little unclear to me whether you're asking whether it's permitted to do work without getting paid for it, or if you're asking whether it's a good idea to do that. I'll address both questions.

Almost any faculty member will be happy to take you on without needing to fund you, provided you have a decent reputation and they have sufficient time to mentor a grad student. I've done this several times when neither I nor the faculty member had funding (or the readiness to commit) by enrolling in an independent study with that advisor. Depending on the institution, this is probably fairly common. You can say you've worked with this faculty member, but you won't be able to say you were an RA with that person because you weren't funded by them. 

Let's move on to the question of whether this is a good idea. Barring any red flags on your financial situation that you haven't mentioned, YES. This is a phenomenal way to get job experience and familiarity with the group. It does make you look like a more dedicated grad student, which should impress your advisor when he/she considers funding you later. It would also give you a head start should you apply for a fellowship. 

I would also urge you to discuss this with the person you're thinking of working with. They might consider funding you if they see that you're already willing to sacrifice (it's a long shot!). They will be better able to advise you than anyone online because they know their funding situation, their field of research, their institution, and the local cost of living. Plus, talking to them early gives you a chance to connect.