How dependent is it on them as a service? If their website/service disappeared off the face of the earth tomorrow, would I still have access to my passwords locally?

I'm still hesitant to use any form of password management that relies on cloud services. I still like Keepass (with auto-updates disabled for security because their updater uses HTTP, of course), for my purposes. I can Sync my keepass file any number of secure ways that don't rely on a single provider.