There are a number of squat variations, and they are all good exercises: front squat, high bar back squat, zercher squats, goblet squats. The mobility issues can be temporary if you work at them intelligently and over time. So I'll address the question in two parts. First is the question you explicitly asked:
By all means substitute front squats in place of back squats. Unless you are planning on competing in power lifting, the back squat isn't absolutely necessary. Additionally, if/when you ever try to re-introduce back squats, do them high bar. The torso is more upright, and like front squats they are also a little more quad dominant. That's OK because deadlifts will take care of your posterior chain.
But don't neglect upper thoracic mobility. This is something I'm working on myself, and my upper back mobility has been preventing me from going very heavy on front squats because the bar wants to keep rolling forward. I found an article on simple thoracic spine mobility with exercises everyone can perform. The one mobility exercise that helped the most would be this one. Also to work on shoulder mobility itself, the DieselCrew has an excellent rehab protocol and warmup routine.
Any time you are benching, overhead pressing, or doing back squats, I recommend starting with the DieselCrew warmup routing to open up the shoulders and getting them ready to work. Use light weights for them, it's meant to be warmup not actual work. After pressing, you might want to work on dumbbell front raises or reverse flies for light weight with high reps (between 10 and 20 reps for 5 sets). You can alternate them on pressing days. The purpose is to strengthen the upper back and rotator cuff so your shoulders return to a more natural orientation.
After a couple months try high bar back squats. Between the shoulder rehab/warmup and the upper thoracic mobility work, your shoulders should now be in a more favorable position rather than slumped forward. Stick with high bar, and use a wide grip. That should allow you to back squat without pain, and it is competition legal should you ever decide to do that.