Currently only Amazon, Google, dropbox, facebook and Evernote are supported for Push Notifications for One Tap logins.
Please allow other 3rd party sites to make use of One Tap Push Notifications. Provide whatever tools are necessary for website developers to integrate LastPass Authenticator push.
e.g. We have a custom website for our business. All of our employees use LastPass Authenticator we would like to add the One Tap functionality to this site.
See this help doc