I don't think that there is one single framework that you can use. In fact, filling the unknowns in the given description would let you spin this into different cases. Let's take two examples that both fit the description:
(a) The non-profit provides sexual violence counselling services in the US. In that case, you'd need to take state law and Title IX into account, as well as potential public funding. This might sound obscure, but any US student applying straight out of college would know to check those things. They would also know that certain states will be more susceptible to the idea and so on. In short, I'd prioritize a PESTLE analysis.
(b) The non-profit organizes exhibitions for promising student artists and invites wealthy members of the public to them, thereby kickstarting the students career. In that case, you'd probably start by identifying which traits make a university a good target. For example, you might go for universities in wealthy towns with an easily mobilized upper class and try to define key members of society who would be willing to head the local chapter of the organization.
I hope that these examples show that there is no single framework you can use - ask clarifying questions and take it from there. Particularly relevant would be how they define their goals - in most cases, it's market penetration or profits, but a non-profit might define its goals very differently.