The issue is extremely complex and it might need a court decision.
To begin with, now that you no longer get a paper ticket most people have no idea about who their issuing carrier (also known as validating carrier) even is. And your issuing carrier is the only one who can help you. (With an emphasis on can, not must, read on.) If you look at a paper ticket you can see the upper left corner have featured the issuing carrier quite prominently. (Image from E-Ticket vs Itinerary vs Booking Reference )
You need the full ticket number which starts with the three digit airline code (then a four digit form number, a six digit serial number, and sometimes a check digit). Most of the time, you can find this in the confirmation email you've gotten:
Then head over to the IATA members list and ... realize you can't search by number so you need to guess it or page through. Thankfully Cathay has a better tool which does allow you to search by number. If that link dies, try this Google search maybe you can find a useful PDF and then you can search inside that. The ticket I posted starts with 182 so now you know that was a MALEV flight from very very long ago (the airline went bankrupt in 2012) because I'd hate to accidentally doxx myself :) Also, as I was picking an email for the screenshot from the flight reservations folder I found sometimes airlines actually do not include this number, bad airline! If you received no e-ticket number, begin to poke around at the Manage reservation section of their website or at worst call them (shudder).
Now you have an issuing carrier. In the best case, this is not the bankrupt airline. Note bankrupt airlines do not necessarily just cease operations, in the USA for example it's almost routine for the large three to enter bankruptcy from time to time. And when that happens, it's rote to include "the Debtors submit that authorization, but not
direction, to continue the Customer Programs in the ordinary course of business and to perform and honor the Customer Obligations as they deem appropriate in their business judgment" in their filing (emphasis mine) so a bankrupt airline have saved you a tour to the court as they already requested to do whatever they please -- and usually receive permission -- from the (bankruptcy) court.
Anyways, let's hope your issuing carrier is not in bankruptcy. If you are even remotely lucky, you actually have a contract with them which they will honor and arrange transport. If you are unlucky enough they will claim they acted as an issuing agent and disclaim all responsibility. Semi-obviously this can only happen if your issuing carrier is not the marketing airline. In this case, your only avenue is to sue them in the relevant jurisdiction (figuring that alone can be a tremendous lot of fun) and then let said court decide whether they were agent-y enough.
Suing anyone but the issuing carrier is usually utterly pointless -- let's not forget we started with the operating carrier being bankrupt -- unless of course your lawyer recommends doing so.
This sort of contrived arrangement happens in two cases, typically: a) more complex itineraries especially when multiple stops are involved b) ... what's a polite word here? ... more interesting booking sites. This whole ball of wax is just another reason to try to stick to the website of an least somewhat relevant airline.