Im hoping to replace this cracked capacitor that looks blown but noticed that the board is peeling.
If I replace the capacitor, do you think the circuit will work properly? Also, can I work on the board (with the power off) while it's still mounted or do I have to unscrew the entire board?
Also any tips on replacing the capacitor would be helpful.
The black part is more likely a current limiting PTC or maybe a VDR overvoltage suppressor than a capacitor. To be sure, you have to analyse the circuit. If the main current path is through the part, it's a PTC.
The degree of PCB damage can't be well seen on the photo (the said "peeling"), so it's to difficult to say if the board can be reworked. Most likely yes.
The 'peeling' looks like conformal coating or some kind of lacquer to keep moisture out. It shouldn't make any difference to performance and can easily be cleaned up then sprayed over again.
I agree with FvM, that isn't a capacitor its some thermal device, most likely a PTC thermistor. It looks like some kind of power supply so it is probably a current limiter.