The card thing is really No Big Deal. It's actually a pretty smart integration with one shortcoming.
Whatever gateway they're using has some support for you maintaining control of the checkout experience while not actually receiving or transmitting card data--a great thing for PCI compliance. So the form is on an HTTPS page on the merchant's site, but instead of posting to the merchant's app, it posts directly to the gateway. The gateway communicates status with the merchant via a synchronous HTTP request or signed data in a redirect URL, and the merchant's software processes the transaction without ever seeing card data.
As the OP says, "they're using HTTPS"--yes, this makes all the difference. The only reason the info appears to be clear text to him is that he's using Tamper Data, a browser extension that lets you play with your own POST data before you send it over HTTP(S).
This is the Right Way to do it for a site that doesn't get some compelling advantage out of building the infrastructure necessary to be PCI compliant. Done this way, PCI compliance requires just the trivial SAQ A. The only problem with this is that either the gateway does not support (sadly, the most common case) or the developer did not implement signing the transaction details on the merchant side.