Haven’t done this in a couple of years, but have definitely purchased directly via Air India’s website and Indigo’s site in the past, and through Cleartrip.

India has been the place that’s been the biggest PITA to use US cards in in general, IME. I think part of the issue is the 3D secure verification they use, and in particular, for me, Chase cards that don’t play that well with whatever version they use in India. I know IRCTC has been a huge PITA ti use a card on in the past for train tickets (I gave up trying eventually and now just go straight to my local Indian Citi debit card), and I forget the site I used to bug movie tickets on where I could never get it to accept it.

may be trying IRCTC tickets again in the next couple of weeks or so…will report here if I can use my cards or not.