Can a customer check the status of their pickup order themselves?

Yes. On your storefront's Track your order page, the customer enters their Order number and Email and clicks Track.

For pickup orders the timeline shows pickup steps instead of shipping ones: Order placed, Preparing, Ready for pickup and Collected.

To keep it accurate, update the order as you go: set it to Ready for Pickup when it is packed (the customer is also notified), then to Collected when they take it. You can see all pickup orders with View pickup orders on Plugins → Click & Collect.