[@Dezuel] I don't think I've ever seen a Rider who didn't have a horse/chariot/something else that they could summon. I agree with your point about not needing to be a Rider to have the riding skill, in fact most knights have decent Riding skills without being in the Rider class (it's actually a Class skill of Sabers as well), though not being in the Rider class does limit what they can do with this skill. But a Rider who doesn't have anything to ride strikes me as odd.

Usually to be summoned in a class you would need to met the prerequisites or that class; you can't be a Caster without being able to cast a few spells, you can't be a Saber without a sword and, I would assume, you can't be a Rider without a mount. That said I'm not sure whether the prerequisite for being a Rider is having a mount or just having a high Riding skill, since I think Lancer Arthoria's profile says something to the effect of "This rank is high enough to have aptitude for the Rider Class" despite not having a mount Noble Phantasm.