If A and B have to decide whether to make a transaction, while C will be harmed if the transaction happens but has no say in whether it happens, that's an externality and market forces do not account for it under any economic model I've ever heard of.
Except with the environment, it's a little murky, because A, B, and C are all affected (perhaps not equally or at the same time, I'll admit). So it's not a "pure" externality at least.
...pretty much all economists agree that a carbon price is the most market-efficient way of doing that...
But what price do you pick? There's no "free market" way to do this. Cap-and-trade will result in a free market price for the available credits or whatever, except the amount of credits is arbitrary. If there was a way for the "market" to determine the available credits, that would be one thing - but there isn't; it's all done by decree. (Kind of a reverse externality if you will - groups A and B decide that this is the level of emissions that's allowed, C's opinion or needs be damned.)
That said, yes, an artificial price on emissions may result in people reducing consumption of those things that emit, depending on the elasticity of demand for those things.