WARNING: A HISTORIAN IS DOING MATH. 

So back in the day, I tried to figure this out by working out the ratio of cavalry to foot in the armies where we have good figures. Here’s what we know:

• Taking Robb’s army at the Twins as our example, we see that 27% of the North’s army was cavalry (6 out of 22k) and 73% was infantry. (You can even go pre-Twins to cancel out the effect of the Freys, as 500 was 10% of the Stark cavalry, meaning that Robb had 5,000 cavalry and 13,000 infantry when he arrived at the Twins). 
• Combining Tywin and Jaime’s armies together, we see that the Lannisters had ~10,500 cavalry out of a total Lannister force of 35,000, which gives us a ratio of 30%.
• Renly’s army at Bitterbridge was 80,000 strong and 20,000 of them were cavalry, which gives us a ratio of 25%. This is part of what started to make me suspicious about Renly’s numbers, because the Reach being the center of chivalry should have a higher percentage than the North, and even if we took 25% as our figure, should have more than 20,000 by itself, and then you have the Stormlands, which should have at least 6,000 cavalry by itself. This is why I’ve come to suspect that a lot of the Reacher lords and Stormlords did not answer the call (at least a third, depending on how much credence you put on Renly’s claim to have 10,000 at Highgarden) and that Renly was, characteristically, overstating his support when talking with Catelyn and Stannis. 

So I would put the ratio of cavalry to infantry at ~27-30%. This would suggest that the numbers look like this:

• The North: 9,450-10,500. 
  
• The Vale: 8,000-10,000.*
• The Westerlands: 12,000-13,500.
• The Crownlands: 4,000-4,500.
• The Riverlands: 5,400-6,000.
• The Reach: 27,000-30,000.
• The Stormlands: 6,750-7,500.
• Dorne: 

  6,750-7,500.

• Ironborn: none.*

* – The Vale might be a special case, in that its army is described pretty much always as just “the knights of the Vale.” So it’s possible they have a higher ratio than normal, which would explain why all these noble houses are in debt. Likewise, the Ironborn have almost no cavalry due to the vagaries of geography, although things were different back when they ruled the Riverlands.

Now, granted, not all cavalry are knights. Northern heavy cavalry are essentially knights when it comes to social class, equipment, training, etc.

even if most of them aren’t given the title.

Squires aren’t knights, but they do fight as knights do on the battlefield, many if not most of them will become knights or are of the social class who become knights. But what about the people who don’t fit that mold?

If we look at Tywin’s army at the Green Fork, we can get a decent picture of how many are which:

The right wing was all cavalry, some four thousand men, heavy with the weight of their armor. More than three quarters of the knights were there, massed together like a great steel fist. Ser Addam Marbrand had the command…

This wing too was all cavalry, but where the right was a mailed fist of knights and heavy lancers, the vanguard was made up of the sweepings of the west: mounted archers in leather jerkins, a swarming mass of undisciplined freeriders and sellswords, fieldhands on plow horses armed with scythes and their fathers’ rusted swords, half-trained boys from the stews of Lannisport … and Tyrion and his mountain clansmen.

So Marbrand’s 4,000 men make up more than 75% of the Westerlander chivalry, which means that Tywin has around 5,333 knights out of 7,500 cavalry – making knights ~70% of his cavalry. If we extrapolate the same percentage across all of Westeros, we would see something like:

• The North: 6,615- 7,350 “sworn swords.”
• The Vale: 5,600-7,000.*
• The Westerlands: 8,400-9,450.
• The Crownlands: 2,800-3,150.
• The Riverlands: 3,780-4,200.
• The Reach: 18,900-21,000.
• The Stormlands: 4,725-5,250.
• Dorne: 4,725-5,250.
• Ironborn: none.*