Why not? Ensuring that the true cost of a product is factored in its price will make people reconsider their purchasing options, and it may potentially show that alternatives are competitively priced.
That does not matter unfortunately as realistically a carbon tax would be below the lower bound estimate of the 'true' cost due to political considerations.
But, even ignoring politics, people have estimated the effects of climate change so it's possible; but there's going to a spectrum of answers.
Cardboard markets are never as good and inventive as the real thing. The companies involved will find a way to not change and compete with the alternatives.
For instance via hard lobbying or even illegal means.
Taxation is not a market-based mechanism. You'd also have to implement prohibitive duties to avoid importing the problem.
Taxation is one way to deal with a shortcoming in markets, namely externalities [1]. The free market is not a panacea. If left to its own designs it leads to all kinds of negative outcomes (such as externalities and monopolies).
