The front squat is a great exercise, but it's less forgiving than squats if you're not mobile or healthy enough in places like your shoulders. Don't miss out on the benefits though -- focus on the basics, like your hand, elbow and bar placement and take it slow. This video explains how.
Alan Thrall starts off by guiding you toward correct hand, elbow and bar placement in a front squat. If your wrists, in particular, aren't mobile enough, it will keep you from holding the bar correctly. Working on mobility takes time, of course, so a quick fix for the meanwhile is using a pair of lifting straps to help.
Thrall also demonstrates a handful of stretches for your wrists and upper back that you could work on as a warmup before every workout.
How to Front Squat with poor mobility [Alan Thrall]