Basically, the position Research Associate I at The Institution (United States) carries the requirement of a bachelor's degree. The application description also provides an email address to lab faculty for those interested in the position. I'm due to graduate in December of this year, so I'm still an undergraduate. Would I be realistic in applying for this position with graduation 4 mo. away, or would it hurt to email the lab faculty and see what they're looking for in the position?
My lab conducts a search for post-baccalaureate fellows every year. In most cases, the applicants are completing their final year of their undergraduate study and have an anticipated graduation date, and are able to share reference letters and a transcript with us. At least in my field (neuroscience) it's completely normal to apply for a post-baccalaureate position before you have officially obtained your degree.