Can the vibrations from the Earth affect gravitational wave detectors?

Summary Yes they can. False positives arising from the acoustic sources you name are ruled out by seismological analysis and the examination of correlation between the separate gravitational wave detection stations.

Pretty obviously, the LIGO detectors are probably the most sensitive microphones ever built. So how do we know they are detecting gravitational waves?

Part of the answer is exquisitely sophisticated vibration isolation engineering. The kind of issues that are addressed by this vibration are explained in the Einstein Telescope design document:

M. Abernathy (& about 30 others), "Einstein gravitational wave Telescope conceptual design study", Published by the Einstein Telescope collaboration comprising the institutions listed in Table 1 of this document, 2011

in chapter 4 "Suspension Systems" and Appendix C.

However, it is impossible to get rid of all the effects of vibration.

So a major part of the experimental design is the fact that there are two LIGO detector stations almost as far apart as one can make them in the United States: one in Louisiana in the South East and one in Washington State in the North West.

Therefore, we look for signals that are present in both interferometers at once. If we accept the relativistic conclusion that no signal can travel faster than $c$, then no the effect on or near to the ground (mining, traffic on roads, and all the human made clatter that we make as a species) that influences one interferometer can possibly influence the other in the time it takes to glean a reading, because those influences travel much, much slower than lightspeed.

This "correlation test" rules most false positives arising from effects other than gravitational waves out. There are then only two possible sources for correlated signals arising at both detector stations with a delay less than 100 milliseconds: (1) acoustic signals arising from a common source within the Earth on the bisecting plane midway between the two stations or (2) from outer space.

Careful and thorough seismometry continuously monitors all seismic waves that arrive at the detector stations. This is enough to rule false positives from source (1) out.

The only other possible source of a signal common to both stations is therefore a disturbance from outer space. When we see such a signal that is absent from the seismometers, we know it is something that modulates the interferometer path lengths simultaneously (or near enough thereto that all other Earthly effects are ruled out) that is coming from outer space.

There are no other known natural effects that can easily explain such correlated detections.

Furthermore, the spectacular gravitational wave event GW170817 was the simultaneous observation of a gravitational wave event by both the LIGO detectors in the United States and the Virgo detector in Italy as well as a gamma ray burst observation within 1.7 seconds by the Fermi telescope. Given that gamma ray bursts are detected by Fermi about once every few days and are observed to arrive as a Poisson process (i.e. they are equally likely to arrive at all times and the statistics conditioned on any event are exactly the same as the unconditioned statistics), the probability of a gamma ray burst within 1.7 seconds of the gravitational wave detection by pure co-incidence is of the order of $10^{-5}$. So GW170817 was a spectacular corroboration of the hypothesis that our gravitational wave detectors are indeed detecting gravitational waves. It is almost certain that the gamma rays and whatever it was that "shook" the LIGO and Virgo detectors in GW170817 was the same source.

Yes they can, and the designers of the LIGO system took extraordinary pains to get the very best isolation numbers they could so as to minimize the influence of external noise on their data. It's worth reading about.