Great question — answering as the UphillTech team:
YouTube Shopping only connects with supported partners such as Shopify, Spreadshop, Spring, and Fourthwall. If you want to handle orders via your own server, you could connect the YouTube store to a Shopify shop and then setup a webhook on Shopify to notify you when an order comes in.
Full answer by Cameron: https://stackoverflow.stackexchange.com/a/79672732 (CC BY-SA 4.0).
Need it done for your business? UphillTech.
