Just do a Google search for "name of university in your area" + "research labs" or "research groups". Look for a lab that you're interested in, and find out who the head of the lab is (Principle Investigator, Director), or find a senior researcher. Get their email address, and go ahead and send your resume with a cover letter.
I can't speak for all research labs, but we did interview people who "cold-called" us.
Alternatively, you can search university job boards, but we didn't always create official job postings every time we needed some one.
As mentioned in another response, universities will prefer to have the work done by students, simply because either they can get them for free, or the process for hiring them is much simpler than hiring an outsider. Research grants often allow for student research assistants but have no budget for external contractors, and if the grant proposal includes that it'll be naturally scrutinized for accountability as the granting institution tries to avoid having funds funnelled to someone's buddy/relative/etc.
What happens in my case is that a project gets seriously delayed because some crucial part needed to complete it was either not planned in the proposal, or the project partner responsible for it failed to deliver. Then someone in the project knows me because we had worked together previously, and recommends me. In some cases where they did have enough budget left I finished that part, the project was reviewed successfully, and everyone was happy. However that's quite intense work, not just the technical part on a tight deadline, but some project partners might be less than optimally collaborative for a variety of reasons.
Academic projects involve quite an amount of politics even if you only care about getting your part to work correctly: you'll need to get some information about the other components and that often conflicts with other partner's interests which are only tenuously connected to the project's technical/scientific success beyond the final review. I wouldn't have known how to deal with it if I hadn't worked for years in different projects as an employee and saw how things play out.
Therefore, from my experience I think your options would be to work in research for a while to make contacts, or contribute to open source research software. In any case your chances of getting market rates for your work are slim.
In spite of all the above, I've had nice experiences doing that kind of work; don't let my comments above discourage you, just keep those issues in mind.
As someone in the comments said, it depends on the grant that the institution that you want to join has or if the lab has some kind of assistant position to fill.
I actually can't help you but I am curious. What kind of academic projects are you looking for? Do you have an specific topic you want to tackle?