For some reason, Microsoft still hasn't provided a method for exporting MS Teams chats.
Turns out there are two workarounds:
Selecting all in the chat and copy/paste.
Opening in a browser and printing the PDF.
In the chat, press CTRL-O (or in the top right click the three dots and choose "Open in new window").
In the new chat window, press CTRL-A to select all the messages.
Press CTRL-C to copy all the messages.
Open a new Microsoft Word document.
Press CTRL-V to paste all the messages.
1. Copy a link to the chat:
2. In Edge > URL bar > paste the link.
3. Use the Teams Web App to see the chat:
4. Using the mouse, drag over the chat messages to select the ones you want to save.
5. Press CTRL-P to open the Print dialog.
6. Save as PDF > Selection Only > Save