Is it possible to provide an option to pick up an order from the physical store in Craft Commerce? We're setting up a Craft Commerce site and the merchant wants to give the option to the customer to pickup the order from their physical store instead of shipping it to a specified address.

If the customer selects that option then they shouldn't have to provide a shipping address or billing address. 

Is this possible in Craft Commerce? If not is there a plugin or a suggested way to do this?
You could make 'pickup from store' a shipping method. 

Then in your checkout form, if a visitor chooses that shipping method, make sure you don't show the shipping address form, and add a hidden input to set order.shippingSameAsBilling to true. Commerce will then populate the shipping address with the billing address details. Code is untested, but it should be something like:

{% if order.shippingMethodHandle == 'pickupInStore' %}
    {{ hiddenInput('shippingSameAsBilling', 1) }}
{% else %}
    {# Show the shipping address form #}
{% endif %}