Can a child get mumps if he had the shot as a baby?

Possible. Even with proper immunizations including boosters, the vaccines do not ever provide 100% protection. However, since most children currently are immunized, the risk of mumps is currently low. There has been a major increase seen in the mid-atlantic area, due to foreign exposures and under immunizations.It is still crucial to immunize.
He could. He should have a booster MMR vaccine at 4-6 yrs of age. But even if he did, no vaccine is 100% protective. Having said that, mumps is rare. Other causes of swollen neck that can look like mumps is lymphadenitis. Talk to your doctor.