Unfortunately, the benefit of the feat Veiled Moon Style, in part, says, “Whenever you move, you can treat up to 10 feet of that movement as teleportation” (Path of War Expanded 86), and Combat on Mounted Combat on Mounts in Combat, in part, says, “Your mount acts on your initiative count as you direct it. You move at its speed, but the mount uses its action to move.”
In other words, when a mount that has aboard a rider that possesses the feat Veiled Moon Style moves, the mount moves and the rider moves, too, but only the rider can treat up to 10 ft. of that movement as teleportation not the rider and the mount.
Even an omen rider harbinger who’s astride her spectral steed can still only apply the benefit of the feat Veiled Moon Style to her and not also to her spectral steed. The supernatural ability spectral steed, in part, says
Whenever an omen rider is mounted and uses a maneuver with a movement component (such as making a charge attack, or taking an extra move action), the omen rider can have her spectral steed make the appropriate move action in her place. (24–5)
…And, sadly, the benefit of the feat Veiled Moon Style just isn’t a maneuver.
(A creature can’t typically attend a second creature so as to count that second creature among the first creature’s possessions. The benefits of attendance are usually limited to objects—see, for example, here.)
Still, it seems legit to allow the rider to employ the benefit of the feat Veiled Moon Style while astride a mount so that the rider alone can count 10 ft. of the mount’s movement as teleportation.
For example, a Medium rider that’s astride her Large mount could enter a typical Medium foe’s threatened area, then—before traveling to the next square past the foe—the rider opts to treat the next 10 ft. of movement that the mount travels as teleportation. She winks out, the now-riderless mount continues moving past the foe, then it moves out of the foe’s threatened area, then, finally, she winks back in once the mount’s beyond the foe’s reach.
In this way, the foe would be unable to make an attack of opportunity against the rider for the not!rider’s 10 ft. movement while not!aboard her mount. (The foe could still attack the mount, though!)
…But it’d probably be okay to allow a rider’s Veiled Moon Style feat to benefit her mount anyway
To be clear, I was really hoping the answer to this question would turn out to be yes because I thought that would be really cool. (I suspect you probably thought so, too, or else you wouldn’t’ve asked.) With that in mind, this reader suspects that in many campaigns a GM could make a house rule that allows the benefits of the feat Veiled Moon Style to extend also to a rider’s mount without endangering game balance. That is, while the Path of War material is awesome and comparatively powerful, mounted combat usually isn’t powerful. In fact, mounted combat is kind of an underdetailed mess (see here and here and even here, for example), so making mounted combat slightly better is kind of okay.
However, this reader strongly recommends against making a broad house rule that says that creatures can attend other creatures in the same way the creatures can attend objects. That will unbalance the game.