All are correct, but there is a subtle difference in meaning:
Without sleeping, we go on to finish our work.
Here, you emphasize the choice or contrast: we could have slept, or we wanted to sleep, but instead, we go on to finish our work.
Without sleep, we go on to finish our work.
Here, there's less of a sense that we could have slept right now; maybe we were up the night before, and now we're tired. This form brings that deficit (without sleep--tired) more into focus.
We unsleepingly go on to finish our work.
Unsleepingly is almost never used, and carries a connotation that we don't need or desire or miss sleep, even if some might think we ought to.
We sleeplessly go on to finish our work.
You didn't mention this one, but this suggests that we would have trouble going to sleep even if we tried; maybe the work is too engaging, or maybe we just have insomnia.