The Fuji GX680 models do not use a conventional mechanical cable release, but they do have a port on the side of the camera to plug in an electronic wired remote shutter release.
These wired remote releases are very simple single button devices that will release the shutter, but that's pretty much it. They came in a short cable version and a very long cable version, but there was no wireless version as far as I'm aware.
Although these electronic shutter releases are fairly nice, for what they are, there are no new ones available now, so you'll have to search for used ones.
You need to be aware that the GX680 III needs a unique remote shutter release, and the remote shutter release for the GX680 I & II will NOT work on the GX680 III. Similarly, the remote shutter release for the GX680 III will NOT work on the GX680 I & II models. Unfortunately, the GX680 III remote shutter release looks pretty much the same as the earlier model that works with the GX680 I & II, and they all use the same plug connector. Fuji only hints at a potential difference by using the III identifier in the label on the GX680 III remote. The label on the GX680 III remote release says "Remote Release III", while the earlier version for the the GX680 I & II just says "Remote Release".
It can be very frustrating if you're not aware of this, since the connector that plugs into the camera is identical for all versions of the GX680 remote release. If you don't have the correct remote release for your camera model, you can plug it in, but it won't work. This would lead you to believe that the remote release is defective (as happened to moi), but it's really just an incompatible remote release. This was a complete boneheaded move by Fuji, similar to the soldered in battery for the I & II backs. They should have used a different plug connector in the III version if it wouldn't work with the older cameras.
Both of these remote releases will plug into any GX680 model, but they will only work in the model they're designed for.
Making Your Own Remote Shutter Release for the GX680
In theory, it should be relatively easy to make a simple electronic shutter release, since you just need a couple of wires, a trip switch, and a connector to plug into the camera. Unfortunately, Fuji used a camera connector that is not commonly used for other electronic devices, and is now out of production. This link GX680 Remote Connector has information about the Hirose RP6 connectors that Fuji used, but it's unlikely you'll be able to find any these days. It's probably going to be far easier, and probably cheaper, to just find a used Fuji GX680 remote, rather than trying to get the correct connector.
But really, who needs the correct connector when it's more fun to just jury-rig up something that will work. The camera RC (remote connection) port has 8 pin holes, but only 2 of these are used to trip the shutter. The trick is to find out what pins are used for each camera model. I've used a multi-meter to determine what pins are active for shutter release on the different remote models. I've labelled the pin holes in the camera body, from 1 to 8, as shown in the photo below. For the GX680 I & II, pin holes 2 and 7 are used for shutter release. For the GX680 III, pin holes 7 and 8 are used for shutter release. You momentarily bridge the two pin holes to activate the shutter.
To make your own wire connections to the camera pin holes, you need some wire that will fit in the holes snugly. I found some bread ties (tin ties, twist ties) that have a thin wire that fits nicely. The wire is probably steel, since it was magnetic. I measured the wire diameter at 0.58mm (or 0.023 inches). This is roughly equivalent to 23 guage electrical wire. I also tried some 24 guage wire and it seemed to fit in well too. I'd look for some 23 or 24 guage solid copper wire for a build your own remote. I'm sure that bread ties come in various wire sizes, so don't assume that any old bread tie will work. Use your imagination to come up with a trigger switch. Anything that provides a momentary closed circuit should work.
How About a Wireless Remote?
You could use a wireless flash trigger to do this. Connect a wireless flash receiver to the camera, and use the flash "Test" button on the transmitter to trip the shutter. For this to work, you obviously need a wireless flash transmitter that has the capability of manually firing the flash (as with a "Test" button). The wireless flash receiver needs to have a PC flash sync port, so you can use a standard PC sync cable. You'll need to butcher the sync cable so it has the PC plug at one end, to attach to the receiver, and cut off the other end so that you can use the two wires to connect to your GX680. I did this successfully with a pair of old PocketWizard flash tranceivers.