I have some experience with this.
No, getting animals to equip armor is not possible without a LOT of weird fiddling around.
The first issue is that you have no way to assign them armor, so they'll never go and pick it up. Even if you had a way to assign them armor, they still wouldn't go pick it up. Only the same creatures as are in your civ in the Entity file will pick up weapons and armor.
There's a way around both those issues, slightly. If you use Dwarf Companion, you can temporarily change the dogs into dwarves, assign them weapons and armor, and they'll go pick them up.
HOWEVER - and this is a big however - creatures are locked at their size upon spawning. So your dog-turned-dwarf is still size 5 and not stout. Thus all the armor you have made is too big for them. And even if you had armor for size 5, non-stout creatures (say you modded in a race that wore such armor and sieged, thus dropping it when you killed them), the weredwarfs wouldn't pick it up, because the game only looks for size 6 stout armor for dwarves to pick up.
There are two ways around this.
The first is a bit ridiculous, and that's making your dogs size 6 and stout. They will then go pick up the armor and wear it, then you can turn them back into dogs. Or you can make dwarves size 5 and remove their stout tag. Neither of those options is particularly good, I think.
The second way also has some consequences, but is a bit more workable. Shields do not have a size, thus the game doesn't worry about checking when you assign something to pick it up. So you can create a shield named "Dog Barding" or something similar, turn your dog into a dwarf, assign it a shield, and it would go grab the dog barding. When you turn it back into a dog, it should still carry it, because once an item is being carried, the game doesn't particularly care that the creature carrying it doesn't have any [GRASP] parts (if you don't believe me, go quickly mod out the hands of your dwarves and restart the game. Your soldiers will now be holding weapons with their lower arms).
However - and again, this is a big one - shields obviously work differently than standard armor. Your dogs do not have [CAN_LEARN], so they would never become more skilled at using shields (though they probably wouldn't get much from regular combat anyway). Because of this, they aren't going to get that much protection out of the shield. Also, your Dwarves will go pick up the "Dog Barding" and use it as a standard shield, and merchant guards will similarly show up carrying them.