I have completed all the missions up to 2-2 The Phantom Mask and collected at least several hundred crowns, but when I go to a vendor at the Haven Bazaar, it displays my crown balance as 0. Is this a glitch, or is there another way I have to get crowns (and if the latter, what is the way)?
You should be able to get every crown you collect. You can check your balance any time by opening Arsenal tab (or you can press I (not L) on the keyboard). It is shown at the bottom of it. But mentioning a vendor, there is little window in it dispaying how many crowns you'll pay/receive for the items you're going to buy/sell. Maybe you simply got confused by it.