The (old) influenza vaccine for kids is pretty safe but its effectiveness sucks, needs two doses at initiation and requires boosters every fall.

Diphtheria and Haemophilus influenzae Type B shots are also pretty useful.

There's not really many shots on here that are particularly suspect, though I can see parents balking at the volume.

The least needed are probably things like the oral Rotavirus vaccine (literally the stomach flu) and Hepatitis B (which doesn't really exist in the wild anymore due to mass vaccination).