I had similar problems with a D70 years ago. It would fail to focus and since it was set to prevent shutter release until focus was obtained - it would not release.
It turned out to be lens contacts that were misaligned in the camera body due to a slightly malformed plastic part, eventually replaced by Nikon under warranty.
In the F mount the contact points all rub over each other every time you mount and dismount a lens. Could dirt or wear and tear be the problem? Having replaced mounts it would seem not so we should look for another cause too.
From a discussion on another forum it appears that certain Canon bodies are very fussy about battery capability. This makes me wonder if you have checked the battery "age" indication in the menu? It could be that your batteries have worn out, or, if using generics, that they are not behaving exactly as an OE battery should. As a battery wears out it can sometimes retain sufficient voltage to appear healthy when at rest but fail to produce enough current when called to do so by an activity such as shutter release.
Lastly Flash, and I think this most likely. Not sure what flash you are using but the Nikon flashes nearly all have thermal protection and will fail to fire when too hot. This happens after repeated shots in a short time period. I don't know whether it manifests in the camera not releasing the shutter or merely the flash refusing to fire.