Prior to introducing the Payburner Wallet as a browser extension, the wallets were (and still are) available as a web app at https://www.payburner.com/wallet/index.html. You can easily move your wallet to the browser extension by importing a the payburner credentials file you export from the web app.
Login to the webapp. Then click on your user avatar, and then on export account file.
Click on EXPORT ACCOUNT. The credentials file should be saved to your Downloads directory and begins with payburner-credentials and ends with json.
Next you will want to install the Payburner browser extension.
When installed click on the round flame icon in your browser bar, then click on IMPORT ACCOUNT
Click on SELECT and choose the file you just saved.
When the account has finished importing, this step is now complete. Your Payburner account is now imported into the browser extension.