Why would the ironborn be poor horse riders and not use cavalry? The iron islands aren’t THAT small, surely?

Horse-breeding takes a lot of land for pasturage. The Iron Islands do not have a lot of land for pasturage, and what soil they have is thin and rocky. 

In addition to the horse-raising problem, you have an issue of opportunity costs. It takes time to learn to be a good horseman, and the best cavalry in world history came from nomadic horse people who rode constantly from extremely early ages. 

The Iron Islands are a seafaring people, and it also takes a lot of time to learn to be a good sailor – so you have to choose one or another to specialize in.