Is It Normal for the Doctor to Hit a Blood Vessel?

There is always a possibility of hitting a blood vessel when inserting a needle into the skin.  A bulging vein is less likely and I would encourage you to call your treating physician to confirm whether or not it appears normal.

Hi Shelly. This is completely normal when having a filler injected around the eyes. The skin in thinner and has more blood vessels in the area, I suggest taking Arnica Forte to speed the healing process. 

It certainly can happen and bruises are the result of bleeding so its not uncommon on the face after injections.  I use a newer technique using a blunt cannula for that area.  This has resulted in less risk of bruising and better outcomes.