First of all, what I'm about to say may vary depending on the field. I'm in mathematics.
The role of the recommendation letters is for someone who can be seen as an authority to tell the admissions committee how well you will do in academia. Someone from outside academia is unlikely to be able to tell them that. You need at least one recommendation letter from someone with experience in academia. Part of the work you do in undergrad, once you realise that you want to go to grad school, is build relationships with professors who could potentially write recommendation letters. A letter from your supervisor at work will tell an admissions committee nothing about how well you will do in academia, unfortunately. You need someone who can say, based on personal experience with grad students, that you will be successful.
Unfortunately for you, it sounds like none of your professors know enough about you to write a good recommendation letter. However, an application (at least in math) without any academic reference would be very unlikely to succeed. I would say that you should pick a professor from a class where you did well, and there was considerable coursework that the professor can base his recommendation on, and email that professor and ask for a recommendation. It won't be an excellent letter, since they don't know you personally, and they may refuse to recommend someone they don't know personally, but you should try. The recommendations are one of the most important parts of a grad school application, and by not getting to know any of your professors well enough that they can write you a recommendation letter you've put yourself at a huge disadvantage.