Can I ask a non-academic for an academic letter of recommendation? A scholarship based on existing mental health problems and conviction to finishing/doing well in school and it requests 2 reference letters and doesn't specify from whom. I'm only a second year in my undergrad, and all of my classes are too big for a connection with a professor. There is no one academic to write me one.

My mom suggested this one family friend who is a nurse and who has known me for 14 years. Is this a good idea? Any ideas who could write the second one?
The purpose of such letters is to get some assurance from "trusted" commentators who can attest to the likelihood of your success in the program. While for most academic letters, professors and teachers are best, in this case it may be different. I can't suggest who would be best, but keep the two elements in mind: 

People who can be immediately trusted to give an honest assessment, and people who can judge your likelihood of success.