I hoping to deliver my twins vaginally. What are my chances of being able to do so?

Discuss w Doc. Different doctors and hospitals have different comfort zones with vaginal twin delivery so it is important to discuss this with your doctor. Some issues that affect delivery: position of babies, difference in weights of babies, weeks of gestation and other medical complications of pregnancy like diabetes and hypertension.
Fair. It will really depend on how the first baby is facing when you go into labor. The first baby needs to be head down in order for you to try for a vaginal delivery.
Position/size. One issue not mentioned by the other docs is the reason we worry.The head is the largest body part to travel through the birth canal and as long as both babies come down head first, there is time for their heads to mold & fit, cord dangling behind. If either the 1st or 2nd go head down, there is the worry that the head will not fit & will cut off the blood supply from the cord in the process.
It depends. There are several variables as to whether twins are delivered vaginally. Typically if the first twin is head down, a vaginal delivery is an option, but it will depend on the skill and experience of the obstetrician delivering you. At times the position of the second twin makes vaginal delivery riskier. It is possible, but is a dynamic situation that will need to be assessed until delivery.