It is hard to be definitive without a picture but micro needle radio frequency my be an option. If the vein is big enough even localized sclerotherapy may be an option.
Under Eye Vein Therapy
The vein you are showing is very small and will not respond to sclerotherapy. The best option will be laser treatment. Even vein specialists will be reluctant to laser it because of its proximity to the eyelid edge but a plastic surgeon who deals with the eyelid (eyelid surgery) and who also does vein procedures (American Board of Venous and Lymphatic Medicine) will tackle this with confidence.
Red Facial Vein Treatment Options
There are two very good options for treating facial spider veins. The first utilizes a pinpoint laser to target the pigment in the blood of these small veins. It is only moderately uncomfortable. The second option involves utilizing radiofrequency energy to ablate these small vessels, of which the VeinWave is a very good option. This device can be used on any color skin as it has no effect on the natural pigment of the skin.
Both treatments may work very well after one session but may require up to three sessions to achieve optimal results. For facial spider veins that do not respond well to either of these therapies, one may consider injection sclerotherapy.