The premise of the question is incorrect. Prior to the Sengoku Jidai, which means most of Japan's "middle ages," warfare was dominated by the samurai, who fought as armoured and mounted horse archers.

Mirroring Europe and Eastern Zhou China, mounted warriors were replaced as the backbone of armies by infantry once the state apparatus was powerful enough, centralized enough, and had enough resources to drill its infantry. Drilled infantry have enough discipline to stand and fight in formation and face the charge of mounted warriors without breaking and running, and of course was still more widely available than mounted warriors in agricultural societies. The newly found discipline and formation fighting allowed infantry to use long polearms, which prevented mounted warriors from charging home. Without the ability to charge home, mounted warriors could no longer break an infantry formation by itself. In Japan, this transition was caused by the hanzei, a system first implemented in the Nanboku-chō in which half of the tax revenue were given to provincial governors in order to both buy their loyalty during the civil war and allow them to mobilize troops to fight it. With it, we began to see provincially raised and organized units appear in the records. The tactical innovation came soon after, in the Hatakeyama succession crisis and the following Ōnin War in which the infantry, now using long polearms, were able to hold their own against the samurai horsemen.

That said, horsemen continued to play a role even in Japanese warfare, where they could be used to quickly exploit faltering infantry ranks or pursue the enemy. And even though there are increasing cases of mounted samurai making conscious decision to fight dismounted, horses were still valuable as mounts to carry their masters around.