You need to have some type of payment module for checkout ...

You could clone the Check/Money Order for Pay on Pickup and make the other two payment modules turn off based on the $this->enabled based on the selected shipping module being Store Pickup ...

One thing to consider is you may want to leave enabled the PayPal payment module, as some customers may wish to pay with their PayPal account, even if they are picking up at the store and it might be easier to manage that online ...