I can't imagine it being Vive.
Gear VR is used because you need very little external hardware. Unless Figment are using some drastically different setup, the headset simply contains a Samsung smartphone (I believe it's the S7 on Galactica?) which deals with all the rendering and playback. The phone contains the screen, sensors, video playback capabilities and the battery. This is why you can by Gear VR and similar products (Daydream, Google Cardboard, etc.) so cheaply if you own a compatible phone. You're just buying a plastic shell with some lenses and occasionally a remote.
They do have a battery onboard each craft to provide additional power and charge the phones for a limited period of time, but I don't think there's much else on the craft itself. The Ghost Train has a power feed to it as it isn't required to travel around a large track, so Vive and the computers are able to have a mains connection.
The little touchscreen panels between each seat probably act as a remote, telling the phone which seat it is on so that it can playback at the right time, and whether to turn on any subtitles or audio description. The phone will then handle the rest, rather than having a video piped into it.
By contrast the Vive is a periferal device. They cost over £700 (consumer rate, obviously Figment/Merlin will be able to buy in bulk at a discount), but then it still has to be connected to a powerful dedicated computer to do all the video rendering. It also (until the wireless kit launches) has to be wired to the PC to draw power, send sensor data, and receive video back. It also requires an external sensor(s) called a Lighthouse, which is what actually tracks your head movement. Vive is, at its simplest, a pair of screens in a headset. There's no real video rendering and playback in the same sense as a phone, and it is not intended to be a mobile device.
Think of them like a tablet (Gear VR) and a normal computer monitor (Vive); they are both screens, but the tablet includes all the hardware to process and render an image. The monitor just shows whatever comes into it from an external source.
Whilst each craft has an onboard 'computer' to monitor seats, etc. don't think of these as computers like you would find in the home. These are usually specialist hardware designed to do a specific job (monitor seats) and will be completely incapable of anything more sophisticated. Duel has a 'computer' in each vehicle, but it's nothing more than sensors to detect position and basic logic to control motors. You wouldn't be able to run Windows and your Microsoft Office apps on them
They just look like a tangle of wires.
Galactica does have a Windows interface on it, as do a few other rides (Rita runs an XP desktop to monitor the launch system), but these are usually for op panel monitors and similar. Safety-critical tasks will be left to specialist systems.
I know that in reality there is a lot more to all of this hardware than just the above, but hopefully it explains the difference between something like Vive and Gear VR
Apparently, according to TowersTimes, the new headsets are the 'Pico Neo CV' VR Headset... Which has the same resolution and refresh rate of the HTC Vive and Oculus Rift!
These could be a possibility as they are completely self-contained devices. They have the playback functionality built right into the headset itself. This may also mean that they have better battery life than the Gear VR setup, as you're not powering the extra hardware of a phone. What's more, if it's using proprietary software and hardware, which it sounds like it is, it should be easier to develop new films on and add additional features.