The answer is, "that depends". If your understanding with the non-profit(s) is that you are coming on board in order to do work in exchange for a reference, then it could be as short as a single project or "enough time for them to get to know you". I find very, very few people give references lightly these days. There are many stories out there about people getting sued for references they gave where the new employee did not do well, so people are far more conservative now. If they are like many of us, they will likely want to work with you long enough to get a good feeling for the caliber of your work and gain comfort that they can refer you in confidence. That could be a month or six months or much more - depends on the person. I think you will save yourself a lot of time and trouble by making it clear that the reference is their quid pro quo for your free work and then discuss what time horizon that will be.

On the flip side, you will want it to be long enough that the reference recipient will take it seriously. This may be your bigger issue. The person checking your reference will likely have your resume in hand, and if they are getting a reference from someone you have only worked with for a month, it will not carry much weight. References gain value with the passage of time. Just how much time is, of course, open individual preference. I like to see someone with references from people they worked with/for over many years. That can be less if the jobs you are applying for are contract/short term in nature.