Hi, I don't have any vaults in my sidebar (as showed here: https://support.1password.com/sidebar/).

You're welcome. There are definitely some features you miss out on by not having the Mac app installed, and also the documentation largely assumes you have the app for your platform installed. If you don't feel like those items are in your way you can of course continue to use the web app exclusively if you'd like.

if I do download the app to my Mac, will it sync up with the passwords I've already entered in just by logging in to a browser?

Correct; you can install the 1Password app on any of your compatible devices and access all of your data that way by signing into your 1Password account.

:)

Ben