Not all Assessor's Offices have online access, but here are the ones that do! 

Richland County:  http://www.richlandonline.com/services/assessorsearch/assessorsearch.asp

Lexington County: http://www.lex-co.com/GIS/Services.html

Kershaw County (taxes only): http://www.kershawcountysctax.com/

Orangeburg County: http://www.orangeburgcounty.org/assessor/DefaultTA.htm

Sumter County: http://www.sumtercountysc.org/

Fairfield County: http://www.emapsplus.com/SCFairfield/maps/

Calhoun County: Available via paid subscription to Orangeburg MLS.

Other Counties not covered by AFS Appraisal Services:

Aiken County: http://www.aikencountysc.gov/DspSvc.cfm?qSvcID=21

Berkeley County: http://www.co.berkeley.sc.us/

Dorchester County: http://www.dorchestercounty.net/OnlineServices.htm

Florence County: http://web.florenceco.org/cgi-bin/ta/tax-inq.cgi

Horry County (pronounced "O-Ree"): http://www.horrycounty.org/depts/finance/assessor.asp

York County: http://www.yorkcountygov.com/Departments/DepartmentsAM/MIS/Divisions/GIS/tabid/231/Default.aspx