Hi, Brittney
It isn’t impossible to get published without social media accounts, but you are definitely making things a lot tougher for yourself – and less appealing for publishers. 
These days so much publisher promotion, and self-promotion, for authors and by authors, is done online. And now, more than ever before, authors are asked to help in that gigantic task of getting the word out – about your book, about you as a writer.  It’s all about networking and having some kind of “platform”. Or how will people hear about you and what your book offers? Best of all, this kind of publicity is free.
If you are serious about a writing career, I would strongly suggest that you do start working on your social media profiles. Pick just one or two options – maybe Instagram and Twitter? – and start making contacts. If you do get a publishing deal you would have to do all this then in any case. So why not embark now, and give yourself the best possible start?
Hope this helps!
Sarah