This is not a complete answer, but I want to point you to the "early music" movement, or what is called "historically-informed performance practice".
When one says "classical" often one is refering to all non-commercial music in the Western European tradition from antiquity to the present day. But to musicians, "classical" refers only to music from a short style period, approximately the years 1750 to 1820.
In the past 40 years or so there has been a resurgent movement called "historically-informed performance" which seeks to understand how musicians sang and played their instruments at the time the music was originally composed (hundreds of years ago), and to try to perform the music in those styles, rather than in the style of singing and instrumental playing that most "classical" musicians are taught in the present day. There are singers who specialize in "historically-informed performance" of music from the early Romantic, Classical, Baroque and Renaissance style periods, and yes, they use a lot less vibrato -- precisely because from the best of our historical research, that is how it was done back then, for several reasons.
On the other side of the coin, however, you won't find singers today attempting to sing with little or no vibrato if they are singing grand operatic music from the era of Wagner or Verdi or since that time, because that music was composed in such a way that a very big, powerful tone and plenty of vibrato are required.
Interestingly many contemporary "historically-informed performance" singers who specialize in Baroque and Renaissance music also sing late-20th-century and early-21st-century classical music (called by many names, including "postmodern" and "avant-garde") because the cleaner, dryer tone with much less vibrato has once again come into fashion for these compositions.
Generally if you are looking for recordings of these singers, you are looking for ensemble productions, such as a recording of an entire opera. These albums tend to be labeled and sorted by the name of the orchestra conductor, not by the individual soloists. Look for conductors, orchestras and groups identified by labels like "early music" and "historically-informed performance" and especially "period instruments". Then identify the singers on these albums and look for more work by these singers.
Here are some examples from my own record collection:
Anything featuring American soprano Teresa Wakim or American tenor Aaron Sheehan.
Handel's Messiah, 1988, conducted by Trevor Pinnock, with the English Concert and Choir. Solo singers are Arleen Auger, Anne Sofie von Otter, Michael Chance, and Howard Crook, and John Tomlinson.
Mozart's The Magic Flute (Die Zauberflöte), 2009, conducted by René Jacobs, with the Akademie für Alte Musik Berlin, solo singers including Daniel Behle, Inga Kalna, Anna Grevelius and Isabelle Druet.