How Can Large Veins Under the Eyes Be Treated?

You need to treat these veins with a great deal of respect. Typically improvements for large visible veins require lower lid bepharoplasty. Injecting these veins can cause very serious complications.
Have a complete evaluation by a board certified plastic surgeon or oculoplastic surgeon.
I hope this helps.

The skin under your eyes is the most delicate of anywhere in the body. Because it is very thin and prone to damage, it’s very common to develop noticeable veins in the area earlier than other place on the body, like the legs. As a board certified dermatologist and laser vein treatment specialist, I often use the CoolTouch Varia to remove unwanted facial spider veins for my patients. The treatment is completely non-invasive and provides excellent results. 

The lower eyelids have a very thin dermis and are prone to early atrophy, development of visible varicose and spider veins, and accumulation of hemosiderin or melanin deposits.
I am frequently asked to treat patients with varicose and spider veins in the lower eyelid. Although the skin of the lower eyelid heals well, I generally do not advise laser treatment for larger and blue varicose veins. The problem is that the overlying skin may hyperpigment, and more importantly, may burn.
Sclerosants are typically not recommended in this area because of the small risk of embolization.
The preferred treatment is laser for smaller capillaries, telangiectasia, and veins. For larger veins, I would advise selective surgical excision of the vein under local anesthesia.