I wasn’t a McNair Scholar, but I was in a very similar program. I also know a lot of McNair Scholars and people who completed similar programs while in undergrad. I’m going to assume you’re referring to PhD programs in public health since you didn’t specify.
The short answer is you can find funding opportunities at many programs. For most fellowships and at most schools, the funding you get isn’t predicated upon any special program you did in undergrad per se. However, McNair Scholars are especially well-poised to compete for fellowship funding because they’ve usually done the kinds of things in college that make someone competitive for fellowships/funding (and honestly, just being in the program often signals to graduate admissions committees “here is a talented candidate from a disadvantaged background.”)
My point being - I don’t think you need to do a special search for fellowships that are “reserved” for McNair Scholars and/or for programs that admit a lot of McNair Scholars on fellowship; if you are an average (or, in your words, above-average) McNair Scholar then you should already be competitive for fellowships in most places you want to attend.
However, your problem is going to be the research experience. What do you mean by “very little”? It was my understanding that McNair Scholars usually completed a certain amount of research experience (10-15 hours) in their junior and senior years of college. Have you done any research in your MPH program?