When you create your first Openledger account, an Openledger wallet will be automatically created as well and stored in your browser.
What this means is, that if you log out of your account, you have to restore your wallet using the backup. If you go to signup, at the bottom there is an option to restore your wallet. You can safely stay logged into your account in your browser, as your wallet should be protected by a strong password.
Yes Browser itself work with strong security to protect your wallet, But there is necessary of back. If in case you lost cockies or cache then it can make trouble so always keep back up at safe place in case of returning your wallet back to access.
That is the worst idea I ever heard of: Storing everything in a BROWSER. Is there no download of openledger ?