Hypothetically, if someone developed a way to capture carbon, is there a market one could sell carbon credits on? If so, where is it and how do you prove you are providing what people are paying for?

Yes - the EU Emissions Trading System (EU ETS) have been running side 2005. Carbon allowances ban be traded on some exchanges - for instance in the UK ICE Europe trades carbon contracts.

If no such market exists, then sell the carbon as fuel until a suitable market comes into being.

How do companies that sell shares on public markets prove they are providing what people are paying for? There are independent firms that audit their financial statements.

There is a company doing both http://www.climeworks.com/