I had this problem from the get-go; I couldn't purchase C-points by using the Steam Wallet option from within STO. I had turned off the Steam in-game overlay when I first installed STO. I turned it back on, restarted the game and was instantly able to purchase C-points. Anyone with this problem may want to double check that you have the in-game overlay enabled for STO.
In case anyone doesn't know how to enable the in-game overlay, right click on Star Trek Online in your Steam games library, go to Properties, then check the box that says something like "Enable Steam in-game overlay (Shift + Tab)".
This worked for me, hopefully it's the answer to everyone's problem with it.