Avanza bus Madrid airport to Salamanca

Take the train (it cost the same and takes the same amount of time more or less) if you can't get the card to work after calling your credit card provider.
You can buy the train ticket on the spot at the airport.
It is probably better to wait to you get to Madrid since there is no guarantee that your flight will arrive in time to catch the bus

I assume KC is trying to avoid having to get from the airport to Chamartin Station to catch the train.

If you have other credit cards, try them. If Renfe allows use of PayPal, do that. If no luck with either of those, call the number on the back of your credit card(s) and tell them you're trying to make a purchase from a Spanish website. I think there's a possibility they may be able to tweak a security setting--but I am not sure.

I cannot guarantee you that you'll find two seats available on the bus you want to take when you get to the pick-up point at the airport. But then again--as already pointed out--you don't know what time you'll be ready to board a bus. It's all a guess. I probably wouldn't spend much money on bus tickets before getting through Immigration at Madrid Barajas. If tickets are dirt-cheap, it might be worth buying them and crossing your fingers. I have no idea how much time to suggest you allow between scheduled landing time and bus departure time. I was in this situation last year with only one practical outbound bus I wasn't sure I could make. I punted by spending the night in a cheap hotel in Madrid.

Thanks for the input! We will keep trying to book the bus...maybe thru a different site.

That's a good idea; don't know why I didn't think to suggest it.