A bunch of websites have live tracking of satellites, e.g. this one. Where are they getting the satellite ephemerides data for all of the iridium satellites?
The ultimate source for almost all freely available orbit data is the U.S. Space Force's https://www.space-track.org/ . It requires registration to create a user account, but is free to use. Services like the one you linked are mainly just repackaging that data into alternate formats, and adding images or different search methods. To minimize the size of the transmission, the original US Government data is distributed not as ephemerides but rather as very short text files (138 characters) called Two-Line Elements (TLE), from which ephemerides can be computed.
Many questions on this site have asked about how to do things with TLEs by hand, but that is a terrible idea. What you need to do instead is go back to space-track.org, download SGP4, and use it to compute ephemerides, and make whatever other calculations you want done to your TLEs. TLEs have lots of drawbacks, but they are free and cover almost everything, so we put up with the problems as best we can.
There is better data out there, in rawer formats, but it is not provided for free to anyone who asks! Operators like Iridium, Inmarsat, and so on make measurements of their own satellites in order to operate them safely, but I don't know of any operator who makes that data available to anyone else. There are a few commercial "space situational awareness" businesses which sell observations from their radars and telescopes, but they are not cheap.