Blood vessels that have thinner walls that are more likely to collapse when blood is withdrawn during a venipuncture are called:

The answer is; veins

Smaller veins are prone to collapsing during venipuncture than larger veins. Due to the suction force created by the needle syringe, the higher outer pressure may cause the walls of the veins to collapse. Once collapsed, the inflamed walls are hard to recover. This stops the flow of blood through the veins. However, the body forms new smaller veins.