No idea who the career NBC-EBC leader is, but this gives some sense of where things stand.
Alden Hill ran for 4745 to be Marlington's career leader.
Duce Johnson ran for 4907 to be Alliance's 2nd leading career rusher. Only E.J. Lilly rushed for more with 5179 career yards, but Alliance was in the Federal League when he played. Lilly may have also held the county single-season rushing mark until Keishaun Sims of Perry shattered it in 2015 when he ran for 3224 (Sims finished with 4810 for his career). Lilly's best single game was 355 yards which I know Duce Johnson surpassed by a healthy margin.
Scott Weber ran for 4891 yards according to the Stark HS icon series the Rep did about 5 or 6 years ago.
I used to have a Minerva game program that listed all of their records, but I must've gotten rid of it. It may not be a career record, and the NBC did not exist yet, but Leon Powell ran for 4307 according to the Icon series. That gives some idea of where Minerva's career rushing leader might stand.
QUARTERBACKS Aidan Longwell, Massillon 6-0, 190, Class of 2020 RESUME: Three-year starter at quarterback for Tigers....
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Louisville's career record is 3469 by Bob Gladieux, so there are no Leps from their NBC days with a high amount of career rushing yards.
What about Dover, Springfield, Salem, South, and West Branch?
EDIT: With regards to Stark County leaders, I totally forgot about Erick Howard. He rushed for 6013 yards during his career at Hoover. Bri'onte Dunn rushed for 5479 yards at GlenOak. Imagine if he had stayed in Alliance...