Firstly, you don't need a degree. I think for the most part now-a-days degrees are fairly useless, unless you're going into a career that you require it, but for programming / development, you don't. That's a big blanket statement and it really depends on who is interviewing you, but personally I've hired people with & without degrees, I never really look for it / at it, it's more about a persons job history & motivation.
That being said, I know you CAN land a remote job straight off the bat without being a grad, however it's not going to be easy, you'd need to get pretty lucky to get someone to buy into you that much, presuming you don't have a significant job history. I don't think I'd personally be very comfortable with it.
I'd try get a few years under your belt working somewhere perhaps locally if you can & then you'll be in a much better position to look for a remote job...