There are three approaches to solving your overall problem - getting the URL manually entered and available to your Action. Two of them tackle it the way you've suggested - involving authenticating to Facebook and tying that to the Assistant account somehow. One solves it entirely inside the Assistant.
Account linking to the Facebook account
You've tagged firebase-authentication, so I'm going to assume that you're using it to do the auth and you've enabled Facebook login through it. This means your user has a "Firebase Account", but they log into that account using Facebook.
I will assume you have a way to get the URL from messenger once they're logged in.
The trick in this case is to setup Account Linking between their Firebase account and their Assistant account. This is done by setting up an OAuth2 server that has access to the Firebase accounts and will create authorization and refresh tokens that are given to the Assistant.
In the Action, you'll send the user to the Sign In helper, which will redirect them to your login page and send back a one-time auth code to the Assistant. The assistant will then use your OAuth2 server to exchange this code for auth and refresh tokens. Periodically it will use the refresh token to get new auth tokens.
When the user returns to the conversation through the Assistant, you'll be handed an auth token and you can use this to lookup the user. Since you also know their Facebook account, you can get the URL via however you planned to do so.
There are drawbacks to this method - it is very complicated, and setting up your own OAuth2 server is not for the faint of heart. You may be able to use something like Auth0 instead of Firebase Authentication to accomplish the same thing, but then you don't have the ease of access to the Firebase database.
Account linking to both Facebook and Google
In your Firebase account, however, you don't need to limit them to just logging into Facebook. You can have them use Firebase to record both the Facebook and Google accounts that they're using. This would "link" the two accounts together in your system.
With this, you don't need to setup an OAuth2 server. Instead, you can have the Assistant use Google Sign In for authentication. If the Google Cloud Project that Firebase is using and the Assistant are using are the same project, then once the user has logged in to your project's web page with their Google account, you'll get an identity token on the Assistant which will contain their Google ID. You can use this to match up with their Firebase account and get the Facebook ID and proceed from there.
But this is still a lot of work and kinda messy, jumping between systems.
Using just the Google Assistant (and maybe a web page)
If you're willing to make some assumptions about the devices your users are using, then you may be able to do it all just using the Assistant. The Assistant doesn't just run on the Google Home and other smart speakers, it also works on most current Android and iOS devices.
So you can detect if they have such a device available and, if they do and they're not currently on it, direct them to switch to that device when you need the URL.
If they don't have such a device available (perhaps because their version of Android is older), and you think this may be a common scenario, you may need to make another entry source available. This could be one of the solutions above, or you may want to just have a simple web page (done via Firebase Hosting and Firebase Functions, perhaps) where they log in using their Google account (so you get their ID) and you let them enter the URL. If you just need a URL - going through Dialogflow may be more complexity than you need.
