This seems unlikely a priori. The Miocene was a period of cooling and grassland expansion, in other words, Horse Heaven. It is from the Pliocene when you see horses becoming displaced by ungulates* like deer and bovids. In fact, a quick glance at Wikipedia indicates that the main victims of this disruption were reptiles in northern regions.
That being said, you could have horses disappearing because whatever at any time. We have discussed previously the extinction of the modern horse species alone in the Pleistocene/Holocene transition (when its range contracted severely, actually) which prompted answers like it'd be replaced in human civilization by donkeys and/or camels. Your timeframe presumes that the entire horse lineage goes extinct well before that, so we wouldn't have donkeys either. Nor onagers, nor zebras. No equids at all.
So, without equids, what evolves to take the horses' niches in the Miocene (reduced severely in the following periods)? Well, you could look at what else was having a golden age in North America in the Miocene along with horses. That else is camel(id)s and distant relatives of them like the
Protoceratids that include the famous
Synthetoceras. This last group was made of grazing, somewhat horse-like runners already. Without horses, do these guys have a shot of surviving the arrival of the ungulates*, coexist with them in the limited roles occupied by modern equids IOTL and migrate west to colonize Afroeurasia (which they didn't IOTL)? Yes? No? Does an entirely new branch of running camels appear and do it instead?
And would any of these guys lend themselves to human domestication and riding? These are entirely different questions. Even with real equids, we do have species that have been domesticated (horse, donkey), and species that were not and are reputed untameable (zebras, onagers). You could end with a different descendant of
Synthetoceras in each of their places and
none of them being domesticable by humans.
*EDIT: I meant to type ruminants.