Generally, yes for plumbing & electrical work you need to get a permit from Franklin County.
For Plumbing,
Contact the Franklin County Public Health Department
See: https://myfcph.org/environmental-health/plumbing/
Or Call (614) 525-3160
Located at;
Franklin County Public Health
280 East Broad Street
Columbus, Ohio 43215-4562
For Electrical,
Contact the Franklin County Building Zoning Department
See: https://development.franklincountyohio.gov/building-permits
Or Call (614) 525-3095
Located at;
PERMIT COUNTER:
142 S. Front St.
Columbus, Ohio 43215