The most important issue is how to give customer confidence that the payment information they provide is protected. From this perspective, chat window is the worst possible option. A person will never feel secure giving out their private data over a chat channel. Event you say it's absolutely secure.
You may add Checkout button to chat window. This button should open a separate screen where user can to provide their payment information. There you may emphasize how the data are protected, e.g. SSL seal, pen-test certification etc. You may keep the chat window on that screen, but it must be visually separated from the payment information delivering a message that payment does not go through the chat.