The incidence of a deep vein clot moving into the lungs from the upper extremities is 1%. What you have injected was a superficial vein - so, the superficial vein would have to extend into the deep veins of the arms and then embolize the heart and lungs - MUCH LESS than 1% - so, don't worry. The plastic surgeon is not well read on treatment of upper extremity varicose veins. Find a board certified vein specialist from the American Board of Venous and Lymphatic Medicine (ABVLM).
Hand veins and arm veins can and are safely treated by both laser and sclerotherapy. The treatment of veins in this area is strictly a cosmetic procedure but can be done extremely safely and with excellent cosmetic results. I have never heard or read of any reporting of clots developing following arm/hand veins sclerotherapy, and certainly, the chances of a clot forming are less than a clot forming in leg veins following this treatment which is very low. I have been treating arm/hand veins for years with both laser and sclerotherapy without ever having a clot (DVT) form. I do not think that you need to worry.