What kind of spider would cause severe vomiting for 36hrs straight and the toe that was bitten to go black/purple?

Recluse. Brown recluse and black widows have a toxic venom that destroys tissue and can lead to gangrene. These need to be addressed as soon as possible to stop the tissue destruction.
See a doctor! Agree with previous answers - however, with a spider bite and these symptoms you should be followed closely. The venom can cause local gangrene and in the toes this becomes a bit more concerning.