Veins on the face can be treated with microphlebectomy, laser or injections.
Microphlebectomy is best for veins that cannot be treated by transcutaneous (through the skin) laser. It involves the physical removal of the vein. There are a variety of tools that can be used for microphlebectomy, and some involve incisions of 1mm, some larger, but typically none larger than 3mm. Typically there is a bruise, but this is short lived.
A 1mm incision typically heals with no obvious scar, especially on the face, which in general heals better than most areas of the body because of an abundant blood supply.
To truly identify which would be best for you, a consultation, or at the very least, a high resolution photo would be needed.
A good rule of thumb, is a vein you can feel, is better for MP, a spider vein, or a threadlike vein, is better treated by injection or laser. The downsides with injection or laser, is several treatments.