Is it normal to feel my baby hiccuping in the womb?

Quite common. Some have speculated that the close position of the babies stomach to the diaphragm causes a rubbing that triggers the hiccups. This is common & generally continues on through the first months after birth. Once the child grows longer & the stomach can drop away from the diaphragm the frequency declines. It bothers mom's more than the baby.
Yes. Babies hiccup a lot! You will feel small quick movements when they happen, especially when your baby is bigger. When we are listening to your baby's heartbeat with a doppler we can hear the hiccups.