When a customer's card is "declined" but you feel it shouldn't be (they are not over their spending limit, the information they entered is correct, etc) the best course of action to take is to have the customer call the card issuing bank (their number should be on the back of the credit card) and inform them that their card is being declined for a transaction they would like to approve.
The card representative should have a list of declined transactions listed by charge amount and merchant name. This will be your own name (not "Chargify"), and is the identifying name you provided to your merchant processor. This is the name that would show up on the customer's credit card statement after a change from your organization.
The representative should then be able to "whitelist" charges from your organization for that customer, at which point the customer can attempt to purchase again.