Is it possible to have varicose veins on your vulva that burst open!?

Unlikely. I agree with dr king it is extremely unlikely that varicose veins on vulva burst. At the time of delivery it could happen. Also they veins could bleed if they were cut by shaving or other grooming.
Doubtful. Varicose veins of the vulva are sometimes seen during pregnancy or afterwards due to valve failure in the ovarian vein. I have never heard of one rupturing (unlike varicose veins of the leg, which do occasionally rupture). They can be treated, usually by an interventional radiologist treating the underlying problem with the ovarian veins.
Not likely. I agree with dr. W-k. It is extremely unlikely. At the time of delivery, it is remotely possible that a vulvar vein could open with the pressure of pushing and the baby's size but, even then, it rarely ever happens. I am not aware of it ever happening outside of the time of delivery.
Yes. In women who have severe varicoses, the veins can burst open.

Related Questions

If I have varicose veins in my legs, is there a chance it will spread to my vulva?

Rarely spreads. Fortunately, varicose veins in the legs do not spread to the vulva. Vulval varicose veins are not directly related to leg veins, though the same genetic tendency that make a person prone to getting leg veins can also predispose you to getting vulval varicose veins. Vulval varicose veins often develop during pregnancy and usually improves once the baby is born.
Minimal. Varicose veins in the legs are due to an underlying vein flow problem in the leg itself, and although associated varicosities can spread proximally, this is very rare. Vulvar or labial varicosities generally arise from a pelvic source, and are most commonly seen in pregnancy, where they can become highly symptomatic, but can be treated very simply, to resolve pain, by serial injection treatments.
The other way around. As dr. Schwartz said, vulvar varicose veins generally come from a pelvic source and most commonly start during pregnancy. From there, they can cause varicose veins in the legs rather than having varicose veins in the legs causing vaulvar varicosities. The good news is that if you do develop vulvar varicose veins with pregnancy, they can be treated easily, safely, and effectively with sclerotherapy.
VV and vulva veins. The cause of varicose veins is malfunctioning (refluxing) of the saphenous system of vein. Although a side branch (pudental) of the saphenous system goes to the vulva, this does not usually lead to vulva VV. Vulva varicosities usually occur during pregnancy and can be due to back up from the pelvic veins and, sometimes, from the saphenous vein. Routine saphenous reflux does not cause vulva VV'S.