How to network with professors when not attending a university

My Story is a long one to tell, so I will just say my current situation and leave out how I got here. I am Interested in studying mathematics but I do not have the opportunity to attend a university due to being hired soon. Recently I learned that I can take college-credit courses without having to enroll in a program. There are two highly reputable schools that offer online upper-level undergraduate math courses: UIUC and Johns Hopkins.

I have no problem investing the money and I foresee no issues on learning the material since I am very motivated and have already studied some of the subjects on my own. The only issue is how to build relationships with professors so that I can apply to graduate schools later in life. Online courses are generally run by instructors who are not doing research. Most graduate schools require 3 letters of recommendation. I'm thinking that investing money in these courses may be pointless if I cannot build relationships with professors Olansgz.

Does anyone know if professors are willing to help someone not affiliated with a university? Should I try emailing professors at UIUC or JHU?