Can I actuate a pneumatic chuck with an air gun? Do I need a check valve to do so?

You don't need a check valve. The chuck is normally closed, meaning it opens when air is applied and closes when the pressure stops. You can use the provided fittings along with an air gun with a rubber tip (7 bar air needed). The rubber tip acts as a temporary seal to prevent damaging the fitting.
Fittings come in a small bag in the shipping package; they do not come pre-installed in the chuck to prevent breakage during shipping.