Just got a tattoo removed with a laser and I am experiencing nausea and throwing up.

This is not a typical allergic reaction to tattoo removal. An allergic reaction is extremely rare and would be swelling, itching, trouble breathing, and drop in blood pressure. I would see your doctor soon to access what is going on.

Patients experience burning pain , tenderness, redness occasionally blistering aftter laser tattoo removal but nausea and vomiting is not expected.
This is also not allergy related which would be itching hives and redness.
I suspect that it is another cause such as a virus you contracted and its exposing itself after the laser treatment and thus you coupled them together. That said speak to your doctor, I would also see if any local anesthetic was used to numb the area. That may cause these symptoms in higher concentrations.