Elle here from Shopify.
When dealing with orders where the customer has selected 'local pick up', you can fulfill and archive the order to indicate that it has been picked up! You can also make a note in the order's timeline for your reference.
If you want all orders to be automatically archived after they have been fulfilled, then enable this option under the Order Processing section of the checkout settings in your Shopify admin.
When fulfilling orders, a shipping confirmation email will be sent; however, for orders set to be picked up, you can opt out of sending the shipping confirmation when fulfilling the order.
There are no additional steps you need to take for handling orders that require local pick up.
Hope this helps! Let me know if you have any other questions.