Is it possible take take part in a research project if I'm not a part of a university?
It's certainly possible, though admittedly somewhat harder outside the framework of a university. Some potential avenues, answering generally - not all of them might apply to your particular circumstance.


Academic/Business partnerships. These are a new hot topic, and in some fields quite active. Universities love them because they're a revenue stream. Businesses like them because its harder to get closer to the cutting edge than at a major university. Look for companies that do this as potential employers? They're good for both dabbling in research, and also as a springboard into the research side of things - I've met several "private sector refugees" in my time.
Research-oriented companies. Quintiles, RTI, Westat, RAND, etc. all come to mind. There are tons of these companies, and many of them both pay quite well and actively publish. Are there any that serve your particular field?
Consulting. Research groups occasionally have funding for outside contractors of one sort or another - and if someone really wants to work with you, they may write such a position into a grant with you in mind. For example, I have some grant support for a freelance programmer. I've known other people hired for a particular expertise, or just "a warm body who isn't a student". This is probably the path you'd end up going down if you both contact your ex-professors and want to get paid.
Volunteering. Academics are cheap. It never hurts to ask if they've got some side project you might be suited for collecting dust in the back.