Can I change a shipping order to a pickup order?

There isn't a systematic way to flip an order from shipping to pickup. You can either do a full refund of the shipping order and reprocess as a pickup order, which would calculate the correct taxes, etc. This is the preferred method to ensure accuracy in accounting.
Alternatively, you can do a partial refund on the shipping order, to only refund the shipping charges to the customer. This doesn't change the order from shipping to pickup, but it does effectively refund the shipping charges.