My organization is hosting several instances of AGOL, and in some cases users are using their logins to authorized their software. We have some users on a difference network domain who are able to authorize without any issues, while users on the other domain are not. Both users are able to access the AGOL sites without issues. What were currently thinking is this could be result of a port being blocked.
Where could I find information on which ports need to be opened to authorize software?
I'm assuming you mean you're running several instances of Enterprise, i.e., a private Portal. Accessing the site itself only depends on whether the web adaptor is accessible.
You should check where your named user licenses are. They may be managed via AGOL, in which case users would need to be signed into AGOL for their software to bu authorized, even if AGOL is not the active portal.
If you've taken your named user licenses off of AGOL, users need to have access to the machine where you License Server Administrator is running. Whether they can do so will depend on that machine's own settings.
Can you confirm where your licenses are currently managed?
Before we can go further, it's important to determine which licensing option was being used in both environment. Based on the description above, it leaning toward Josh's assumption that Pro is using a Named User licensing option from on-premise portal as opposed to AGOL. Verify by doing the following to make sure we're on the same page:
1. Working environment - Start Pro and go to Setting>Licensing tab. If using a Named User licensing option, it should list the portal URL in the Settings section. You can also click on the "Configure your licensing option" to get to the Configure Authorization license dialogue. Verify the License Type is set to Named User License. Which portal is selected, ArcGIS Online or ArcGIS Enterprise?
2. Failed environment - Start Pro to the login dialogue. Click on the Configure Licensing option to get to the Configure Authorization dialogue. Again verify the setting is the same as the working environment.
If you are using Named User from ArcGIS Enterprise like Josh mentioned, I can provide further suggestions on how to configure the license manager to work with the firewall.