The problem with the vertical lines around the lips is that regular lipstick will bleed up into those lines. Permanent make-up will help solve that problem. There are a few things that can be done to improve those lines prior to the permanent make-up procedure which will make it easier for the tech.
Consider a CO2 laser resurfacing procedure around the mouth area. This will substantially smooth those lines. Tissue filler, like Restylane or Juvederm, will temporarily smooth the lines. The permanent make-up technician should have no problem implanting the pigment even without pre-smoothing the lines but the final result will be more pleasing if those lines are addressed.
I would also recommend you select a technician who works with a board certified dermatologist or plastic surgeon to ensure a safe experience and a quality result!