Flight search APIs are heavily locked down to either require payment, or in the case of free public search tools, to verify that they’re being accessed “through the front door” by clicking on the UI.
ChatGPT will generally just hallucinate flight options if you ask it to make a search, or it will give the options it can find in Google web search results.
Unless someone builds a specialised tool with paid access to GDS and airline APIs, the technology that will make searching flights with AI possible is called “computer use”, ie the model interacts with software running on a computer to perform a task, like opening Google flights and clicking on fields and entering text. This technology is still in its infancy but you can already try it with Anthropic or ChatGPT desktop. It doesn’t do a good job yet.