I don't know what sites most people are using but Expedia has a 24-hour cancellation window, even for non-refundable tickets, I occasionally found myself using Expedia to generate a booking and then cancel immediately once done. I don't think there is any need to bother your TA anymore in this day and age when flexi e-tickets are a thing.
Why Expedia you ask? Zero FX risk. You are charged / refunded in your currency of choice.