I am slowly rolling out a script that impacts sales orders (adds a catalog item after submit) and do not have a sandbox 😅. For this script to work with web orders (Site Builder) and Farapp orders would it just be as simple as adding "web application" and "Web Store" to the execution record or am I missing some major steps?

If all of the logic is the same no matter the context then adding the execution context on the script deployment should be fine. If you have to do branching logic, you'll have to check the context first.