John,

The answer to your question will depend on the type of testing.  If you are just looking for hardness, iron, and those types of tests, Any water treatment company (selling water softeners and home treatment equipment)  can perform those tests.  If you are looking for coliform bacteria, nitrate, or chemical toxins then a certified analytical lab would be your choice.  The Ohio Department of Health has a list of certified labs at this link.  https://epa.ohio.gov/static/Portals/28/documents/labcert/Combined-Lab-List.pdf

There are other analytical labs that are not on this list that could also provide the services you need.  There is not a lab in Licking County but there are options in Franklin County which would be close to you.