I'm actually very familiar with the Dragonlance series as a whole, so I'll give you my perspective on this based on the source material being the Dragonlance series as a whole. To base it off of numbers, I'll use percentages of armies so you can easily incorporate it into your games.
Black Dragons (5% to 7.5%) - moderately rare, confined mainly to swamp areas. They do serve in the dragonflight armies, although their presence is not as prominant as blues.
Red Dragons (5% to 7.5%) - Moderately rare, mainly independent breeds. Some reds are present in dragonflights, roughly to the same degree blacks are.
Green Dragons (0.01%) - Treacherous and vile, the greens in the books are not people friendly or team oriented. They don't work with the armies of Takhisis, and don't make for very good mounts due to their breath weapon being a chlorine gas that would murder it's own rider.
White Dragons (0%) - These are mentioned in the books but they're relegated to frozen areas and depicted as the most cowardly and small of the chromatic dragons. I keep checking the books but I can't find a single example of one seeing battle anywhere, so I would say a white would have to be of an unusual personality and unique backstory in order to be played.
Blue Dragons (80 - 85%) - These are the primary mounts of the evil armies. The blues are very, very similar to the Metallic dragons in that they are loyal and battle honourably. This is most prominently displayed in the post War of the Lance books where the entire planet unites to battle Chaos.
How many soldiers would you mount on dragons for a standing army? Approximately 10% to 15% of the standing army. This means if your cities are large, with an army of 3,000, then approximately 400 - 450 of them would be on dragons. From that, you would just apply percentages from above to get your final tally.
Assume 400 dragon riders. 320 would be blue, with the remaining 80 split between black and red. If a player wants a green or a white, I personally would need them to have a very compelling reason for that as it doesn't really fit the lore.
Unlike the chromatics, the metallic dragons were evenly represented across the board. Because of this, I would recommend just doing a straight 20% split across all 5 breeds, Copper, Bronze, Brass, Silver and Gold. The number of dragons on the good side would be moderately less than the evil side (I would go with around 60% to 80% of the total evil dragons). This is based on the fact that the Draconians significantly reduced their numbers when they stole the eggs, the dragonlances were required to even the score, and the post Chaos war era with the dragon overlords almost totally obliterated them.
So this also depends on the era you choose to go with. I've broken down the presence of the dragons in the separate eras below.
Pre-Huma era - dragons equal
Huma era - dragons abundant on both sides
War of the Lance - Chromatic abundant, Metallic scarce, Dragonlance makes Metallic overpowered
Chaos War - Chromatic and Metallic in moderate amounts (due to prior war helping even the numbers), Chaos dragons overwhelmingly abundant and extremely powerful
Mina era - Chromatic dragon overlords present (with flights of chromatic dragons that are aggressive to other chromatics of differing colours), Chromatic dragons overwhelming but slaves and fiercly territorial, Metallic dragons almost extinct and in hiding.
Now you've asked for post War of the Lance. Contrary to other answers, the chromatics were NOT obliterated. They withdrew after Berem and his sister were reunited and the portal to the abyss closed. The chromatic dragonflights were still abundant as is evidenced by the systematic annihilation of the armies of good just prior to the Chaos war. This was a period just one generation after the War of the Lance as it dealt with the children of the Heroes of the Lance, including Kitiara and Sturm's son Steel Brightblade. Considering that the only thing that held the armies of darkness in check prior was their own infighting, when they instituted a knighthood in much the same manner as the Sword, Crown and Rose, they handily obliterated the good guys. With all things being equal, this is a clear indication of superior numbers at play.
So post War of the Lance you'd have a fairly even playing field for numbers, with superiority going to the good dragons with dragonlance riders. If you're twenty years past War of the Lance, this pendulum has swung to superior numbers in the chromatics again.
WotC forums has a guy who crunched some numbers for population, source thread here:
Based on that, I would assume a 1000:1 ratio for humanoids to dragons to get a good, solid number. That will give you about 4,000 dragons to work with, of which I gave a sample randomization below that seems to fit with the story lines fairly well.
For metallic: I randomized 211 copper, 220 brass, 352 bronze, 521 silver, and 296 gold totaling 1,600 dragons.
For chromatic: I randomized 1,152 blue, 535 red, 481 black, 140 green and 72 white totaling 2,400 dragons.
This supports multiple armies of several thousand personnel outfitted with dragonflights of a couple hundred mounted riders per side. Since battles on that scale are limited to two or three standing armies at a time, that fits in with the lore fairly well while keeping things in perspective of post War of the Lance figures.