Not really what I asked. This talks about speed, but no indication they marched in step.
I am sorry you think that. You asked if they marched in step and Vegetius indicates that they did - they marched in the Roman Military Step.
If want a specific answer it does help to be precise. What you asked was vague as it covers some 2000 years of history from the founding of Rome to the fall of Constantinople. Neither did you specify whether you were asking about drill on a parade ground. marching on a route march or tactically in battle. From what you now say I think you actually meant "cadenced step".
There is an astonishing number of books, academic papers in journals, Doctorate theses in University libraries, larger reference libraries and some are available online. Some are free some charge a fee. They cover Roman drill, marching songs, military music, pay, religion, marriage, recruitment, training and almost every aspect of the Roman Army (and navy). I found one on the colour of horses in Roman Britain. In addition the Romans (not all were military men) wrote a lot about military operations though no actual drill manuals have survived they are referred to in many other sources.
A good starting point is "The Complete Roman Army" by Adrian Goldsworthy (one the best authorities on the Roman Army) which has a very useful bibliography.
Happy research! You will find the answer and also gain a great insight into the fascinating Roman military and the many and varied opinions of the academics.