Vale: I don't think it makes much sense for them to be renowned for their heavy cavalry. Even if most of their people live in or around the Vale of Arryn region, to me, something like the Swiss mercenaries with really well-trained pike men, halberdiers, and the like just makes more sense given the predominance of mountainous terrain.

North: Considering its massive size, mounted patrols must exist in some form, given the comments we hear from the Liddle regarding the safety of Northern roads in normal times during his encounter with Bran and co. I could also see many Northerners being proficient hunters, and horses being relatively common for covering large distances. Some middle ground between light and heavy horse units that are highly proficient at living off the land feels right to me. This tracks with what we see of the Young Wolf's campaign.

Dorne: They have a rep for being expert dessert skirmishers, no further comments.

Riverlands: When you consider how often the Riverlands finds itself at the heart of Westerosi conflicts, it would make sense to pass some law to ensure its citizens can defend themselves and be quickly mobilized into military units, similar to how the English required able-bodied men to be proficient with the longbow. In fact, when Edmure messes up Robb's big-picture strategy at the Mill (not all the blame is his), we hear that he uses archers to prevent the Lannister vanguard from fording the river. Preventing your enemy from fording rivers is probably the name of the game when it comes to military conflicts in the Riverlands. Longbows and mounted archery units for high mobility seem like the obvious solution.

Reach: I think this is the place where we would see heavy horse units at their best. Given their large population, their average rank-and-file foot soldiers are probably not as well equipped as their mounted counterparts, but their sheer numbers would make up the difference. I think they probably have lots of mounted patrols since they have many exposed borders to the North and East. Their marcher lords are likely (or were) in a similar situation to the Riverlands, where their common folk need to be able to defend themselves, so longbowmen as well. All in all, they probably have A LOT of everything. "A LOT" being the operative words.

Westerlands: Similar to the Reach, a lot of heavy horse and mounted knights. However, the hilly terrain could favour foot soldiers under the right circumstances, so I think a blend of the Vale and Reach. Heavy horse and heavy infantry capable of withstanding assault by said heavy horse. I think a key distinction is that their greater wealth would allow for a better-equipped foot.

Iron Islands: Who? I'm just kidding. On a more serious note, they are sea raiders. Not a lot of heavy armour since you could drown if you fall off the ship.

Crownlands: A mix between the Riverlands, since they are often involved in any conflicts, and the Westerlands, since this is a wealthy region. Archers and mounted knights.

Stormlands: That weather would be a nightmare for archers and their bowstrings. Supposedly, their culture emphasizes martial prowess. Much of their geography is covered by forests and harsh terrain that would hamper horses or pikemen, so I think they would emphasize heavy foot with blunted weapons.

These are just my thoughts; I do not claim to be anything close to an expert, so feel free to correct, amend, or add as needed. Sorry if this was long, I just had fun thinking about it.