I think you are looking at this the wrong way. Rather then ask is it possible to do it, the real question is why didn't we domestic the horse earlier.
If you can figure out the limiting factors and then explain in your story/world how your society overcomes them, then it is only logical that domestication would take place early.
At the highest level someone would domesticate an animal when the benefit of having the animal is greater then the cost involved with keeping them. The cost is easy to figure out. Whatever time or energy it take to care for the horse as opposed. A modern 1000 pound horse needs 15-20 pounds of hay a day, or an equivalent amount of forage. Modern horses are bigger now, so maybe you can get away with 10 pounds per horse per day.
Let's also say you have 3 horses, a dam(mother), a sire(father) and their offspring, realistically you'd likely need more then this for an effective breeding program, but it's the minimum you would need to be able to support.
So your cost question becomes, where do I get 30 pounds of plant matter per day. In the spring through fall maybe it isn't too hard, but that is a lot of effort and time on your part during the winter. Time and effort you aren't spending finding your own food.
Now the benefit side of the equation. What do I gain by having an essentially wild horse? I am not going to ride a wild horse, and as others have said we don't have wheels so a chariot does not provide a solution.
In our time line we hadn't started farming yet so even using the horse to pull a plough is out. At that time we mostly interacted with the horse as a source of food to hunt, so maybe you make the argument they first were do domesticated as a food source, but why horses and not something with less upkeep and a quicker generational cycle, like rabbits.
We know that domestication of the horse is possible given the right conditions, so you just need to find a way to write the story that those conditions are met. Maybe farming started early in your world or maybe your tribe lives in a valley that stays warm all year round because not hot springs, or maybe they make some annual journey to a place with no food, and the easiest way to eat is to bring the meat in as live horse and slaughter as you need the food.
Point being just find a way to get the cost benefit ratio to point the right direction for long enough and you can get your domestic horse early, which could then be trained into the Calvary you are looking for.