Women who find out that they are expecting twins have a lot to work through, from how to have a healthy pregnancy, to how to parent two newborns at once, and how to give birth to twins. Although the cesarean section rate for twins is large, giving birth to twins vaginally and naturally (however you define the concept of "natural birth") is possible for many women... although your circumstances have to be right, as does your healthcare provider. More than half of all twins are still born vaginally in the US, dispelling the myth that being pregnant with twins automatically means a cesarean section.
The spectrum of possible risks is indeed higher with twins, something that will be reflected both in the prenatal care that twin mothers to-be receive and the monitoring and procedures during labor and birth. If you are pregnant with twins, and you would like to give birth vaginally, that will mostly be possible if your babies are both vertex (head down), or baby A is vertex and baby B is breech. If baby A is a breech,or both babies are, giving birth vaginally will be more risky and women will also find it harder to locate a healthcare provider who will oversee their birth.
In case one or both of the babies is in a transverse (side lying), a vaginal delivery is not possible, and a c-section is necessary to facilitate the birth. What if you are not comfortable with the idea of epidural anesthesia, that may be a bigger issue. Many hospitals have a policy that all twins should be born in the operating room to make a speedy c-section possible as soon as it may be needed.
"Mandatory" epidurals can also be part of the policy that make emergency cesareans easy and quick. Those women who already have epidurals only need the dose to be turned up before a c-section, and don't need epidural placement or general anesthesia. If you really don't want an epidural, that is certainly understandable. Talk to your doctor about it, but do think about the possibility that you will need a general anesthetic if you do need a c-section, in which case you would not be awake during your babies' birth.