In the end, I could only get it to work with this code:

Set copy_range = Worksheets("Purchase Requisition") _
..Range(cell_range.Offset(cell_count, 0), cell_range.Offset(cell_count, 7))
Range("A4:H4").Copy
copy_range.Select
ActiveSheet.Paste

It seemed to want the entire range, from one end to the other, as opposed to
just the first cell I wanted to paste it to, like the Macro had.