This is an area of the face which rightfully provokes much anxiety and concern. Injecting medicine into blood vessels around the eye? Couldn't that cause blindness?
Realizing that the anatomy of the periorbital tissues is separate and distinct from the eye is the first factor to consider. Sclerotherapy works wonderfully for vessels that are the proper size for this technique (small, like spider veins on your leg, but big enough to get a 30 gauge needle into). This includes facial or periorbital veins. Though some of my colleagues express concern about risks around the eye, this is generally because of lack of experience with sclerotherapy in this area, and naturally recommending a laser or procedure that falls within their own training.
I have done sclerotherapy for 30 years, and was extremely cautious when I first injected vessels around the eye. I can happily tell you that this is not only possible, but works quite nicely, just as with leg spider veins! The key difference with facial or periorbital vessels in particular is that they are somewhat more difficult to immobilize and accurately stick the needle into. Once that is accomplished, however, the vessel is gone. I have taught laser surgery for over 20 years, and can tell you that lasers work best only for the extremely tiny vessels of rosacea, port wine stain, or sun damage, and would not be applicable for larger vessels like you are describing.
Suture or stripping techniques are "overkill" in my opinion, though certainly effective, more costly, and leaving tiny but more noticeable scars than sclerotherapy.