I want congressional, state legislative district, or county VEP turnout rates
The best proxy for congressional or state legislative district voting-eligible population turnout rates is citizen voting-age population turnout rates, which can be obtained from recent American Community Survey data. ACS extracts of citizen voting-age population estimates by racial and ethnic groups for many geographic levels including congressional and state legislative districts are available here. Historical data for congressional districts can be obtained from Census Bureau reports.
CVAP is the best proxy for VEP since there is no easy method of apportioning correctional populations to districts. For interested persons, Prisoners of the Census identified prison populations in the 2010 census group quarters data at the census block level, but I have not attempted to compile these data for districts.