A quick scout of other forums and the general internet seems that people have tried to book with Linksair and are not able to do so because of this issue with them not being able to take payments. It's very easy for a company to set up an card payment facility, unless they are not credit worthy.
Afraid to say I can see this being a complete non starter. What kind of company launches a website with a schedule of flights no one can book? I'd be very surprised if the flights operate at all.