Can I use an e-cigarette during my tattoo removal process?

In short. . .Yes. The laser breaks the ink particles up and then your lymphatic system helps clear the ink out of the area. Because of this, tattoos above the knee (arms and neck are the best) clear faster. Also, massaging the area in between treatments helps increase clearance, especially for tattoos on the feet or ankles. Keep the sun off the area. Smoking can impeded lymphatic flow by decreasing vascular flow. The affect on tattoo removal is more noticeable in older people and tattoos below the knee. Vaping can cause some of the same vascular problems as regular cigarettes, but I have lots of patients who smoke and things are going fine.

No.
The longer answer: Evidence has shown that cigarettes hinder ink clearance, presumably by the effects of nicotine on circulation. It's a perfectly reasonable mechanism, because nicotine's ill effects on other organs (besides skin) are well known. Using this reasoning, any source of nicotine--even a patch--would be detrimental to your tattoo removal success. It should all be avoided, except for proper, temporary use of nicotine patches for smoking cessation.
Replace nicotine with regular exercise, and you're set!