Thanks for posting @gingerwraps If you enable the option for local pickup your information will be displayed at some point during the checkout process. The store template, Order Online, will show a popup box upon entering the site, or when clicking on the store page. If you change the template to Shop All then the store address will show on the checkout panel if the user chooses pickup.
There isn't a way to remove or hide this information at this time. If you do not have a physical address that you feel comfortable making public, you could not offer the standard pickup option, and instead mark your items as shipping fulfillment. When you create the shipping rule (under Fulfillment > Shipping) you can title the rule "pick up", then send the pickup address in the order confirmation email or let the buyer know that you will be making pickup arrangements after purchase.
This isn't a perfect work-around, and may not work well for you since you have less control over who can request a pickup. I recommend making sure to notate somewhere on the site how your pickup will work i.e. coverage area, day/time restrictions, etc. Perhaps as a popup notification, or included on the item description page.
If you want to continue to use the pickup option and the pickup features you will need to use a physical address. Perhaps you can reach out to your local chamber of commerce to see if they have any suggestions on a public address that can be used as a placeholder. Then follow the previous mentioned steps to provide the valid information in the order confirmation email.
