ChatGPT for Legacy iOS works by directly using your API key from your organization to authenticate with OpenAI's API in order to have permission to use the endpoints and models. That API key is needed and as of right now, there can not be another way of logging in. The API key which is inputted within the application's first launch sequence or the Settings page is stored within the application's preference file in it's local storage. Additionally to that the user's E-Mail and username is stored there aswell for general profile customization reasons.
After the first launch sequence, ChatGPT for Legacy iOS downloads the users avatar in order for the user avatar to be displayed with accuracy to the user. This avatar is stores in the local temporary directory and is, as usual, always deleted whenever caches are removed with something like iCleaner.
All outgoing apiary requests are going to https://api.openai.com. ChatGPT for Legacy iOS uses modern TLSv1.3 SSL OpenSSL libraries which do not rely on iOS's native networking libraries but cURL instead. This makes the network traffic more secure and allows the application to still work despite it having updated cipher suites.
Application updates are regularly checked whenever the app launches. For that, it contacts chatgptlegacy.com and processes it's response.
If you would like to check for the legitimacy of my claims, here's the repository for ChatGPT for Legacy iOS, hosted under the GNU General Public License v3.0.