There is good information on the foreign born population of every county in the United States (and for that matter, every census tract), which is broken down by age as well.
There is good information on the number of persons naturalized in each state going back more or less indefinitely, which when compared to the number of foreign born people in each state can be used to determine the percentage of foreign born adults in each state who are not U.S. citizens, and extrapolated to the county level foreign born statistics to get a good approximation of the number of adult non-citizens in each county in the U.S. Just under half of foreign born persons are naturalized U.S. citizens. About 7% of the U.S. population consists of foreign born non-citizens, while 6% of the U.S. population consists of naturalized citizens of the United States.
As the map indicates, for the vast majority of counties in the United States, the foreign born population is very low. Only one in twenty of non-citizens in the U.S. live outside major metropolitan areas, while one in six native born citizens do.
And, one can use survey data and more isolated efforts to determine these figures to estimate a credible percentage of non-citizens who are registered to vote or do vote.
"As of 2015, the five counties with the largest foreign-born populations (Los Angeles County, Calif.; Miami-Dade County, Fla.; Cook County, Ill.; Harris County, Texas and Queens County, N.Y) account for 19% of the national immigrant population in the U.S." Source (citizen and non-citizen alike). Outside greater DC, greater NYC, Detroit, Florida, Texas, Nevada and the Pacific States, and a few other states in the Southwest, the maximum number of non-citizen voters even if an absurdly high (relative to reality) 5% adult non-citizens were voters, is negligible.
Also, since most of the concern about non-citizen voters is ultimately driven by the concern of conservatives that non-citizen voters will flip elections for Democrats, it is worth noting that most counties with lots of non-citizen adults are also overwhelmingly Democratic by margins that far exceed the highest imaginable percentage of non-citizen voters in any reality based analysis, and another significant percentage of those counties are very safe Republican leaning counties, where again, non-citizen voters wouldn't make a difference in outcomes.
The number of counties where non-citizen voting could conceivably make a difference so as to justify a comprehensive study of the type suggested in the original post, is very small, and predominantly limited to Florida. So, part of the reason that such studies are rare is because the justification for them is so weak.