If I do not have PBS Passport, will I still be able to watch shows on PBS.org and my station's site?

Yes. Many videos can be streamed online at PBS.org and other sites and apps without requiring a Passport benefit. And of course, PBS shows will continue to be broadcast to the public over the air. 

PBS Passport is a membership benefit that allows you extended access to some shows when their streaming rights have expired or they have otherwise become unavailable for general streaming.