Yes, having crossbows as weapons would work*. Four across the back, firing forward, as air to air missiles. Four across the chest, firing down, as air to surface missiles. Each one is connected by a wire to one of the pilot's fingers. While that digit is curled, the wire is loose. Straighten the finger, the wire is pulled taut and the connected crossbow fires.
Low tech automatic crossbows exist. The Mythbusters built one (though at ballista size). They have a gravity loading magazine, and a crank to pull the string back between shots. You could have the crank mechanism be tied to the pilot rolling or waving their wrist.
An alternative (or maybe an addition) would be to have a sail connected to the bowstring. After the crossbow is fired, the sail is pulled taut, and the force of the wind on the sail pulls (or at least helps to pull) the bowstring back.
What a drag*
Here's the thing. The crossbows you could sell. There are some fairly light but powerful crossbows out there. Crossbows fired by tripwires and pressure plates are moderately well known (you could use the tripwire example as a lampshade).
The reloading? Not so much. Gravity fed magazines stick up and out by design. They add a lot of drag and more weight, neither of which is good for a wingsuit. Plus if the pilot needs to dive or roll, the gravity feeder won't work, or since they are touchy as is, might get out of alignment.
On top of that, neither of the reloading methods are especially believable for a wingsuit. Neither one is going to provide the force to cock a reasonably deadly crossbow, at least without adding a lot of weight or drag. Which, as mentioned before, is bad.
Crossbows as weapons without needing hands, yes. Reloading, not so much.