I have some pretty noticeable dark vein like mark near one side of my mouth. Is it due to some deficiency? Any removal option?

There are normal veins that do occur around the mouth area.  I would discuss your problem with a plastic surgeon or vein surgeon to evaluate whether you have a cosmetic or medical vein problem. Raffy Karamanoukian MD FACSLos Angeles

Veins on the face and around the corners of the mouth are rather common, certainly not uncommon. They can be painful in certain situations and should be treated.Options of treatment are topical lasers depending on size and depth, IPL, thermocagulation, RF or sclerotherapy.An in person examination is important for full evaluation.Contact a facial plastic surgeon, dermatologist or vein specialist. 

Facial veins are very common and can occur on one side and not the other.  There are several options to treat these veins including sclerotherapy, laser, IPL, Vein Wave, Vein Gogh and fine needle radiofrequency .  There are no natural products to remove these veins.  You should see a vein specialist who treats facial veins for an evaluation.