Camera lights up and all seems well, but on firing shutter, camera shuts down and display turns black

Since it's a film camera, there are a couple of things to check:
1. You have left the lens cap on.
2. If this is a single lens reflex camera, take the lens off and make sure that the mirror isn't stuck in the "up" position. If it is, GENTLY pull it down until it stays down. At this point, your shutter may be open. Try cocking the shutter again, fire off a test shot, and see if the mirror sticks in the up position. If it does, you probably need to replace the mirror bumper foam - age can make foam deteriorate into a gooey, tar-like substance that can make the mirror stick in the up position.
3. Does the camera have a T (for Time) setting on the shutter speed dial? If so, turn it off the Time setting and set it to a manual shutter speed such as 1/60 - or whatever your 'X' synch speed is.
4. Your batteries died in mid-exposure. Replace the batteries. You may have enough juice in them to show the viewfinder display, but not enough to finish the exposure and close the shutter.

First try pressing the DISP or DISPLAY button on the back of your camera. If that doesn't help, try taking a picture with the screen black. Is the picture also black? If so, a stuck shutter is another common failure mode for digital cameras. The symptoms of a stuck or "sticky" shutter are very similar to CCD image sensor failure. The camera may take black pictures (for shutter stuck closed), or the pictures may be very bright and overexposed, sometimes with lines, especially when taken outdoors (for shutter stuck open). To confirm a stuck shutter, put the camera in any mode other than "Auto", and turn the flash OFF (you don't want to blind yourself for the next step). Next look down the lens and take a picture. You should see a tiny flicker in the center of the lens as the shutter opens and closes. If no movement is seen, then you likely have a stuck shutter. If so, please see this link for further info and a simple fix that may help.

And here you go, this should help. A stuck shutter is another common failure mode for digital cameras. The symptoms of a stuck or "sticky" shutter are very similar to CCD image sensor failure. The camera may take black pictures (for shutter stuck closed), or the pictures may be very bright and overexposed, sometimes with lines, especially when taken outdoors (for shutter stuck open). To confirm a stuck shutter, put the camera in any mode other than "Auto", and turn the flash OFF (you don't want to blind yourself for the next step). Next look down the lens and take a picture. You should see a tiny flicker in the center of the lens as the shutter opens and closes. If no movement is seen, then you likely have a stuck shutter. If so, please see this link for further info and a simple fix that may help.

When you take off the lens and fire off a shot, does the mirror go up and return or does it stay up? If not, is it possible that it is staying up when the lens is connected? When you fire a shot, can you hear it returning?

Do you mean that no picture file was actually recorded, or do you really mean that the resulting picture file recorded a black picture?
If you mean that you're camera is taking black pictures, this sounds like a stuck shutter. A stuck shutter is a common failure mode for digital cameras, and it seems to be especially so for any of the Canon A6XX series (although these cameras are awesome otherwise). The symptoms of a stuck or "sticky" shutter are very similar to CCD image sensor failure. The camera may take black pictures (for shutter stuck closed), or the pictures may be very bright and overexposed, especially when taken outdoors (for shutter stuck open). To confirm a stuck shutter, put the camera in any mode other than "Auto", and turn the flash OFF (you don't want to blind yourself for the next step). Next look down the lens and take a picture. You should see a tiny flicker in the center of the lens as the shutter opens and closes. If no movement is seen, then you likely have a stuck shutter. If so, please see this link for further info and a simple fix that may help.

A stuck shutter is another common failure mode for digital cameras. The symptoms of a stuck or "sticky" shutter are very similar to CCD image sensor failure. The camera may take black pictures (for shutter stuck closed), or the pictures may be very bright and overexposed, especially when taken outdoors (for shutter stuck open). To confirm a stuck shutter, put the camera in any mode other than "Auto", and turn the flash OFF (you don't want to blind yourself for the next step). Next look down the lens and take a picture. You should see a tiny flicker in the center of the lens as the shutter opens and closes. If no movement is seen, then you likely have a stuck shutter. If so, please see this link for further info and a simple fix that may help.

its actually probably the display. there is 3 different displays. 1. shows picture with icons 2. shows just the picture and 3. is a solid black screen. just hit the display button to cycle through until your desired display is present

Jim, it really sounds like one of threethings. Either will require a repair shop.
1. the film sensor is not recognizing film in the film chamber, or film across the back, therefore not telling the advance motor to advance film...
2. The film advance motor itself needs replaced.
3. The rear door latch switch needs replaced.
Since everything else seems to do OK, I doubt the main computer board is bad.

Are the pictures also coming out black? If so, a stuck shutter is another common failure mode for digital cameras. The symptoms of a stuck or "sticky" shutter are very similar to CCD image sensor failure. The camera may take black pictures (for shutter stuck closed), or the pictures may be very bright and overexposed, especially when taken outdoors (for shutter stuck open).

To confirm a stuck shutter, put the camera in any mode other than "Auto", and turn the flash OFF (you don't want to blind yourself for the next step). Next look down the lens and take a picture. You should see a tiny flicker in the center of the lens as the shutter opens and closes. If no movement is seen, then you likely have a stuck shutter. If so, please see the following for further info and a simple fix that may help:

http://camerarepair.blogspot.com/2009/05/simple-fix-for-stuck-shutter.html

I was cleaning my CCD and damaged the shutter. Same problem. Click to take a picture and the camera locks up and powers off. Turn the camera back on and the shutter releases and the camera resets and powers on. $281 service call later and Precision fixed it quickly. Replaced the shutter. They warranty their work for 6 months. Priced way too high, but nobody would touch it. As soon as I told other repair services how I messed it up, they turned me down. I had to much money invested in the camera to toss it in the closet.