Yes, tattoos can be covered up with makeup and airbrushed. However, it makes the job more difficult and time consuming for the makeup artist and there is a very real risk that the makeup applied to the tattoo(s) will rub off on the clothing which is a huge no-no, especially in the modeling world!
Tattoos can be partially or completely removed by laser removal and intense pulsed light therapy. These procedures are very expensive (think thousands of dollars) and usually require 1-10 sessions not to mention the very real risk of scarring!
Some famous models have tattoos (most got their tattoos after they were famous though). There are agencies that specialize in tattooed models but they are few in numbers. Tattoos’ are permanent and will hinder your career!