Full Disclosure: I'm a beta premium backer of the game
You can only purchase ships with in game currency. At least, as of right now that's the way the game works.
Most of the early backers, as well as kickstarter supporters, will get a free second ship when the game launches, but it's an Eagle, a bit of a sidegrade to the default Sidewinder. The only other advantages for people who paid early access is the ability to play the game sooner, which means more practice before things launch, and a better insurance rate, which reduces the amount of in game currency to replace your ship after it blows up.
There are micro-transactions for paint jobs for the ships you fly in the game, but it's purely cosmetic.