Dear Mr. McKinney,
Thank you for posting your request on History Hub!
According to the Carter County Website there appears to be no portal to look up court records online unless it’s part of a docket: http://www.cartercountycircuitcourt.com/gensessions.asp. There is a phone number of 423-542-1835 in which you can call directly. Some of the county records may have been sent to the Tennessee State Library and Archives. That agencies web address is: https://sos.tn.gov/tsla.
We hope this is helpful. Best of luck with your research!