First off, either Tristana's cannon is not .357, or she is insanely small (.357 is just over a third of an inch bore (roughly 9mm), scaling based on that, she's less than a foot tall). Last I checked, Yordles are about 3 feet tall, and her cannon is about 2/5 her height, which would make it a much larger size (in caliber, that would be 14.4 roughly, which is just over a foot in diameter in the bore).
Graves shotgun is probably at least 8 gauge given model scale (fun fact, lower gauge means larger bore), while Urgot's knees appear to most likely be some kind of vent, not an actual shotgun.
Cait's gun should be somewhere around a .308 or .30-06, based both on scale and the fact that it's a hunting rifle (.30-06 is the highest standard hunting round for things not considered 'big game'), but she has an ability called '90 Caliber Net', which would imply that it's a .90 (or higher) caliber gun (such a thing does exist, search '.950 JDJ' on Google and you can find info (and a lot of funny Youtube videos)), which would imply she's using a hunting rifle almost twice the caliber of standard anti-material rifles (most such rifles are either .50 BMG, or 14.5x114mm (which is .57 caliber)), and implies she has absolutely insane upper body strength or that the gun itself has some kind of magical recoil compensation system.
Jinx's minigun is most likely either a .223/5.56x45mm (the same as the US M16 rifle, M4 carbine, and M249 light-machine gun) or .308/7.62x51mm (the same as the US M14 rifle, the European FAL, and the US M60 machine gun). Anything much larger caliber than that would be too hard to manage the recoil for (unless like Caitlyn's rifle it has some kind of magical recoil suppression system). Her rocket launcher appears to be a lightweight recoilless rifle with an internal counterweight.
Gangplank's pistol is a filntlock or caplock styled after a smoothbore single-shot pistol typically found in the times of Blackbeard (the smoothbore aspect is further reinforced by the short range of his Q, and the single-shot aspect is reinforced by the long CD). Given model size, I would guess it's probably about .52 caliber (most such pistols were designed to fire musket balls, which were typically just over .50 caliber).
Fortune appears to be wielding a pair of dueling pistols with a design based on a blunderbuss (she is very obviously not wielding a pair of blunderbuss, as they were essentially primitive shotguns, and fortune is very obviously firing single projectiles). Similar to gangplank, they appear to be flintlocks or caplocks, though they have to have some kind of magical reloading system, as the visible design would only allow for a single projectile. As far as what caliber, my guess would be at least .50, probably much higher (back in the times of black-powder firearms, it was not uncommon to find handguns with very large bores).