How fragile are mirrorless cameras in terms of vibration and bumps while in a case in a backpack while mountain biking?

Probably less of an issue with mirrorless cameras than with DSLRs or SLRs because they have fewer moving parts (no mirror). In my opinion (having torture tested my Fujis over the last few years with no failures) if you carry the camera in a minimally padded case and don't drop it or abuse it you'll have no problems. The x100 series are probably a little more robust than their mirrorless DSLR kin because of the fixed lens and their light weight.

It seems logical that a mirrorless camera would be, if anything, LESS subject to damage from vibration and bumps. But my experience of cameras is that they're surprisingly robust. Have you found different?

I've dropped a Z Cam E2 from shoulder height on a wood floor on it's back. I was not able to detect any damage. It's been running flawlessly for months. The E2 is built like a tank. But the reality is, there are few if any moving parts on these cameras. If you're interested in a rugged camera, look at the build materials of the chassis. Some are made of polymers but others are machined from metal. You will want to find models made with a metal chassis and that are weather sealed. They're not water proof but they can take the damp and the dirt.

A high-grade digital camera like the Fujifilm X100F is very sturdy and would have no trouble handling vibrations.

Hiking is fairly typical outdoor activity and a padded case should soften the bumps enough to avoid damage. In fact, a mirrorless camera is more sturdy than a DSLR since it has one fewer moving part. Some cameras that take things further even lack a shutter which is what Nikon did in their AW1 which is waterproof and shockproof to drops of 2 meters. There are also compact cameras built like that which can handle much rougher handling but for hiking, the construction of the X100F is sufficient.

Modern cameras, both mirrorless and DSLRs, are pretty robust. Some of the top tier professional models are designed to take punishment day in and day out when used by imaging professionals. There are even old jokes that have been around seemingly forever about how they are so sturdy that a Canon 1-Series with an "L" lens could be used to beat someone to death and the camera would still be usable to take a photo of the victim. (Reporters who see a lot of tragedy tend to develop a bit of gallows humor, just like the policemen, firefighters, and soldiers whose work they cover.)

But even lower end models are fairly durable. The parts of most cameras that are most vulnerable to bumps and bangs are the lenses, and there's no real difference in that respect between MILCs and DSLRs. Both types of systems have some lenses that are legendary for being built like a tank. Both have some lenses that have a reputation for losing optical alignment at the slightest amount of impact. So it's really an individual lens-by-lens and body-by-body situation.

If you keep your camera in a well padded case and carry that case on your body, it's doubtful that you'll have any problems with vibration or impact while the camera is in the bag. If your camera is mounted on a tripod and falls over the edge of a cliff you'll probably have major issues no matter what kind of camera it is. It's not so much the fall that causes the damage, it's the sudden stop at the bottom.

It is true that mirrorless cameras tend to have fewer moving parts than SLRs and DSLRs. But the technology of SLRs (including DSLRs) is decades more mature and many important lessons have been learned along the way in the 100+ years that reflex mirrors have been part of cameras.

Mirrorless interchangeable lens cameras (MILCs) are relatively new on the scene. Some of the technologies inside them are even newer as camera designers find more ways to leverage the advantages of a mirrorless design.

Roger Cicala, the founder and all around technical guru at lensrentals.com, has published a blog entry that shows he has found a number of MILCs with IBIS (in body image stabilization) that are prone to developing cracks in the parts that connect the sensor to the IBIS mechanism. These cracks result in measurable amounts of variation with regard to the flange to sensor distance, which is critical for a lens to be able to focus to infinity. As Roger points out in the blog entry, it's a very low percentage of certain models that have been found with the issue. But with the number of units Roger's company deals with on a daily/weekly/monthly basis, he has a large enough sampling size to say that it is not just an isolated camera here or there.

As we've already observed, there can be more difference on a camera-by-camera and lens-by-lens basis than may be found between on particular MILC and another particular DSLR.