If I set Search Engine Visibility to No, can shoppers still visit my store?

Yes. Search Engine Visibility only affects search engines. Your store stays open, and anyone with the link can browse and buy as normal.

With it set to No:

• robots.txt asks search engines to stay away from every page
• Your homepage, products and other main pages also tell search engines not to index them

It doesn't hide the store from people. To close the store to visitors, use Settings → Web Settings → Maintenance Mode instead.

The setting is in Settings → Web Settings → General Settings. Click Submit after changing it.