I was a little worried my description might be a little confusing.

How, where exactly should I put the Selection.EntireRow part?

This part checks to make sure the user has selected only a cell in column C. If that is not what is selected, the nothing happens when they click the OK button.
    With ActiveSheet
        'Make sure they are in the Part Number column
        If Intersect(ActiveCell, .Columns(3)) Is Nothing Then Exit Sub
That's exactly the part I'm having trouble with.

How do I get from that selected cell to using the same row, column S to enter the number? And that same row Columns A - E to shade and apply the border?