A child obsessed with horses usually cannot have one at the party. The games that work are the ones that let them act like one instead.
A broom handle with a sock horse-head, or a stick with a paper horse-head taped on, made by each child beforehand, turns a simple relay race into the whole afternoon's highlight. Making the hobby horse is itself an activity, not just prop-building.
A short visit to a stable, a farm with ponies, or even a single friendly horse over a fence does more for a horse-obsessed child than any game you could invent. It is worth the extra planning if one exists nearby.
Grooming a soft toy horse, "mucking out" a hay-filled tray, or feeding pretend carrots gives children who love horses something to do with them, which is closer to what actually interests them than riding alone.
Yes — the hobby-horse race and stable-job games do not depend on one. A real visit is a bonus, not a requirement.
From about three, once running with an object in hand is manageable. Keep the race short and the course simple.
Part of our guide to Birthday party for one obsession.