Here is the Apollo 11 (Columbia and Eagle) graph of center of gravity during the AS-506 flight, taken from the APOLLO/SATURN V POSTFLIGHT TRAJECTORY - AS-506 (rather large scanned PDF):
I selected Apollo 11 flight for historical significance, but you could find many other postflight telemetry and flight analysis documents online for various launch vehicles, or approximate it yourself using e.g. such calculations (and technical data of specific launch vehicle configurations that can often be found online, say on Wikipedia):
Determining Center of Gravity - cg. Image source: NASA Rocket Fundamentals
Where weight $w= m \times g$ (mass times gravitational constant) and $d$ is distance from reference location.
Do note though that center of gravity does depend on individual vehicle's specific configuration, payload mass distribution, flight profile, that it will shift vertically during flight as stages consume propellants and during staging, and perhaps more significantly that this data is on its own only perhaps relevant to static stability of the launch vehicle. For anything else, i.e. dynamic / flight stability, you'll at the minimum also require values for center of pressure as a function of time.
Here's a graph showing movement of both along the long axis for the first 140 seconds of a Saturn V flight, taken from Description and Performance of the Saturn Launch Vehicle's Navigation, Guidance, and Control System, Walter Haeussermann, MSFC 1970 (PDF):
Variations of center of pressure and center of mass during flight.
You're right though that typically CG only moves along the long axis (vertically) and that the mass distribution of launch vehicles and their payloads are balanced to remain centered along this axis. For launch vehicles, mass distribution and center of gravity would most commonly be established during so-called boilerplate mass simulators, and for their payloads also during payload integration. Oftentimes, ballasts weights (or even secondary piggyback payloads - like e.g. OSCAR satellites) would be used to assure that launch vehicles and payloads are spin stable.