The problem here lies in your if statement - you are always storing the newly entered codes in cells C1 and the date in D1. You need to dynamically work out the next available row number and use that instead. Try something like this:
Public Sub DataInput()
...
If FoundCell Is Nothing Then
MsgBox SearchTarget & " was not found."
Dim nextFreeRow As Integer
nextFreeRow = Range("C" & Rows.Count).End(xlUp).Row + 1
Range("C" & nextFreeRow).Value = InputBox("code?")
Range("D" & nextFreeRow).Value = Now()
Else
...
End If
...
End Sub