The Left Vein on my Neck Becomes Visible when I Am Talking, Can it Be Removed? (photo)

This neck vein is part of normal anatomy, and shouldn't be removed.  Hopefully you can try to accept it.

Thank you for posting a picture.That is the external jugular vein (EJV). The external jugular vein is usually inversely proportional in size to the other veins in the neck (i.e., internal jugular vein) and occasionally double their size.The external jugular vein drains into the subclavian vein lateral to the junction of the subclavian vein and the internal jugular vein.The EJV is not a cosmetic vein and should NOT be removed or closed. 

You have a mildly prominent external jugular vein that is visible when you speak. Pick a favorite female singer and watch them hit the high notes. All the veins stick out on their neck becasue of incressed venous pressure with the air you exhale. This normal. One could have it removed but it would be hard to find a surgeon to do it. The scar would be worse. Hopefully, you can try to accept that is normal. Check out Gwen Stefani or Mariah Carey.