stLinkCriteria needs the fieldname criteria should be applied to. Also, variables should be concatenated and the combobox reference is a variable. Don't put the combobox reference within quotes, concatenate, otherwise you just have the literal string 'Me![Store_Name_Lookup]' instead of the value of the combobox. I don't bother creating VBA variables when they are used only once.
Code:
Dim stDocName As String
Select Case [Store_Name_Lookup]
Case "Cartridge World Canterbury"
stDocName = "Rpt_Canterbury_T_C"
Case "Sittingbourne"
stDocName = "Rpt_Sittingbourne_T_C"
Case "Strood"
stDocName = "Rpt_Strood_T_C"
Case "Chatham"
stDocName = "Rpt_Chatham_T_C"
Case "Maidstone"
stDocName = "Rpt_Maidstone_T_C"
End Select
DoCmd.OpenReport stDocName, acViewReport, , "field name here='" & Me![Store_Name_Lookup] & "'"
Note use of apostrophe delimiters in the WHERE CONDITION argument. Dates use # and numbers nothing.
Are these 5 reports identical in structure? If so, why 5 reports? Why not just 1 and use the WHERE CONDITION argument? If each report is set up for a specific store, why do you need to use the WHERE CONDITION argument?