Latin is a language; the culture is Roman. When you say "Latin" today, virtually everyone assumes you mean Latin America. You'd do better to Google "music of Rome" or "Ancient Roman music," or something like that. I found a musical group named "Synaulia" on Wikipedia. They claim to recreate the music of Rome. But as far as I can tell it's all instrumental; there's no singing.
I think you'll be disappointed in your search. The Greeks left us pretty good information about their music, including descriptions of their modalities, the mathematics of their scales, and even a few bits of what we call "sheet music." The Romans did nothing like that. Most of what we think we know about Roman music is implied from their practice of copying Greek culture. We don't have any of their melodies or lyrics so we can't sing their songs.
The early Christian church was scandalized by theatrical music so it was suppressed--a phenomenon which unfortunately still survives in some "modern" Abrahamic communities. I didn't spend a lot of time looking at Synaulia, but I'm sure all they're doing is playing music of Greek form, using the instruments the Romans are known to have had.
There is of course a mountain of songs with Latin lyrics, but they were all written in the Middle Ages by people for whom Latin was no longer a living vernacular language. Mostly hymns and other religious stuff, by then the church leaders realized that you can't expect to be popular by denying music to people. But nobody was inspired to write love songs, drinking chants or war ballads in Latin.
About 75 years ago Carl Orff decided to remedy this. He searched diligently and found a collection of silly, profane, and otherwise totally un-religious poems written in the 13th century by students and other iconoclasts. He set some of them to music, for a choir and orchestra, and named the composition "Carmina Burana," the title of the poetry collection. He did his best to make the music sound ancient and the result is very entertaining. The last time we went to an Ozzy Osbourne concert they played selections from Carmina Burana for intermission music; it fits right into a heavy metal mood.
A friend of mine is a composer and concert pianist. The local German-American association recently put on a performance of Carmina Burana using three choirs and a stripped-down instrumental ensemble of two pianos plus a head-banging percussion section. They held it in a very elegant and acoustically perfect theater. It was quite a memorable evening.
