Do the orbits of satellite galaxies show the influence of dark matter? I understand how galaxy rotation curves show the influence of dark matter, or something else that would produce similar behaviour, such as Modified Newtonian Dynamics (MOND). Do the orbits of satellite galaxies show the same influence? 

In MOND terminology, the internal gravity of the satellite galaxy is not relevant to this question. The question would be if the gravitational effect of the host galaxy on the satellite was Newtonian or MOND?
The short answer is "Yes, the orbits of satellite galaxies definitely show the influence of dark matter (or something like MOND, if you prefer)." For galaxies outside the Local Group, you can only measure their radial velocities from Doppler shifts (i.e., velocities along our line of sight to the galaxy), but you can still look at the distribution of these velocities for satellites around some central galaxy and work out the approximate amount of mass required for the satellites to remain bound -- and thus you need dark matter. (The first hints of dark matter, going back in the 1930s, were from a larger-scale version of this: Zwicky and others measured the radial velocities of galaxies in galaxy clusters and found they were too high to be explained by the visible mass of the galaxies in the clusters.[*])

For satellites of our own galaxy, it has become possible, in the last twenty-five years or so, to measure proper motions (motions across the sky, perpendicular to the line of sight) as well as their radial velocities and so we can measure their actual 3D space velocities. This paper by Fritz et al. (2020) uses proper motions from Gaia observations for 45 satellite galaxies to estimate the mass of the Milky Way's dark-matter halo. They find this to be about $1.5 \times 10^{12} M_{\odot}$, consistent with other measurements.[**] (Note that unlike the case for measurements of gas clouds orbiting in a disk -- the original data indicating dark matter in spiral galaxies -- you can't assume the orbits of satellites are circular, so the calculations involved are more complicated, but they still give roughly the same answers.)

[*] Later observations with X-ray satellites demonstrated that galaxy clusters contain significant amounts of normal baryonic matter in between the galaxies, in the form of very hot, dilute, X-ray-emitting gas. However, even when you add this to the stellar mass in the galaxies, you still don't have enough mass in the cluster to explain the radial velocities.

[**] This value can be compared with a baryonic mass for the Milky Way of about $5 \times 10^{10} M_{\odot}$ in the form of stars, plus maybe another $1 \times 10^{10} M_{\odot}$ in the form of gas.