Modifying Existing Code to Check if the entry exists

As Mordred suggests you can compare the variable to the cell before populating it.

You can also reduce the duplication in your code by simply assigning the reference to the correct worksheet before transferring values.

I have assumed if the value already exists then no information is transferred

With Me
    
    Set ws = Nothing
    
    Select Case .h9.Value
    Case "ACTIVE"
        Set ws = Worksheets("Active_Data")
    Case "INACTIVE"
        Set ws = Worksheets("Inactive_Data")
    End Select

    If ws Is Nothing Then
        ' unexpected value for h9
    Else
        iRow = ws.Cells(ws.Rows.Count, 1).End(xlUp).Offset(1, 0).Row
         
        If ws.Cells(iRow, 3).Value = fn Then
            MsgBox "Cell already set"
        Else
            ws.Cells(iRow, 1) = .h1
            ws.Cells(iRow, 2).Value = .h2

            ws.Cells(iRow, 3).Value = fn
            Me.h3.Text = fn
             
            ws.Cells(iRow, 4).Value = .h4
            ws.Cells(iRow, 5).Value = .h5
            ws.Cells(iRow, 6).Value = .h6
             
            If notify.Value = "True" Then
                ws.Cells(iRow, 7).Value = "Yes"
            Else
                ws.Cells(iRow, 7).Value = "No"
            End If
             
            If remind.Value = "True" Then
                ws.Cells(iRow, 8).Value = "Yes"
            Else
                ws.Cells(iRow, 8).Value = "No"
            End If
             
            ws.Cells(iRow, 9).Value = .h7
            ws.Cells(iRow, 10).Value = .h8
            ws.Cells(iRow, 11).Value = .h9
            ws.Cells(iRow, 12).Value = .h10
            ws.Cells(iRow, 13).Value = .h11
            ws.Cells(iRow, 14).Value = .h12
            ws.Cells(iRow, 15).Value = .h13
             
            ws.Cells(iRow, 16).Value = Now
            Me.h14.Value = Now
             
            MsgBox "The record has been sucessfully added!" _
            , vbInformation + vbOKOnly, "Record added"
        End If
    End If
End With