For Near-Earth Orbits (and sometimes farther), telemetry data known as Two Line Elements(TLEs) are commonly used to propagate orbits. Propagating an orbit is a term that means predicting future locations from a known state. TLEs look like:

HST                     
1 20580U 90037B   22010.54856767  .00000893  00000-0  42745-4 0  9993
2 20580  28.4696 134.5382 0002786 129.9204 289.6315 15.09931360542325

Hubble Space Telescope (HST)

There are a number of places to look up recent TLE sets on almost anything orbiting Earth.

A searchable SATellite CATalog:
https://celestrak.com/satcat/search.php

TLE’s are valid for propagating orbits with decent precision for a few days after their original Epoch.