I am an American living in Korea, and I bought a flight to Singapore with a layover in Ho Chi Minh City through Kayak/Kiwidotcom. When I arrived at the airport I was not allowed on board because, according to the clerk, I needed a transit visa. I looked it up, spoke to people from a Vietnam visa organization, and they said I didn't. The clerk said this airline in particular, Vietjet, didn't offer transfers within the airport, so I'd need to check back in, i.e. leave the gates and go through immigration again. I wasn't refunded the ticket and I'm still obviously angry about the situation because how was I supposed to know that this one airline doesn't offer transfers. I bought another flight, but I want to know if there is any way I can argue this to get at least a partial refund, either through the airline, Kayak, or my credit card?
VietJet is a LCC (Low cost carrier) and like most LCCs is a point to point carrier. You check in for A to B, then at B claim your bags and check in for B to C.
For domestic to domestic flights this is not a big deal, but for international to international it does mean clearing immigration and customs, to claim and recheck your bags.
But sadly as has been mentioned in numerous similar questions, it is the traveler's responsibility to make sure they have the correct documents for the trip they are undertaking, which includes any transit points. Kayak is not responsible for telling you, nor is VietJet. The airlines only responsibility is to the government of the destination to make sure a traveler does have the necessary documents before boarding them and denying them if they don't (as in your case).
You can try filing a complaint with your credit card company, but they will likely side with the airline, as they understand where the responsibility lies.