Hi A22. We commonly treat the veins under the eyes and all areas of the face with a long pulsed yag laser. The two that we own are the Sciton Profile and the Laserscope Lyra. What's more important than the brand of the laser is the type (long pulsed yag) and also the operator. Just because the practitioner is using the right type of laser does not mean they will be able to clear the veins. Find someone that does this regularly and can show you photos of their own patients to back it up.
You can find before and after photos of facial vein removal in Los Angeles at the link below. Good luck.