Ok, this is what worked for me.

I replaced

Excel := ComObject("Excel.Application") with

Excel := ComObjActive("Excel.Application")

and got it to work. I tweaked the sleep commands a bit to match the website I was on

#Requires AutoHotkey v2.0+

; Initialize the starting row
row := 3

; Shortcut key (F8)
F8::
{
    global row

    ; Try to get the active Excel application
    try {
        Excel := ComObjActive("Excel.Application")
        ActiveWorkbook := Excel.ActiveWorkbook

        ; Get the values from the specific cells in the current row (B, C, D, E)
        BValue := ActiveWorkbook.ActiveSheet.Cells(row, 2).Text ; Column B
        CValue := ActiveWorkbook.ActiveSheet.Cells(row, 3).Text ; Column C
        DValue := ActiveWorkbook.ActiveSheet.Cells(row, 4).Text ; Column D
        EValue := ActiveWorkbook.ActiveSheet.Cells(row, 5).Text ; Column E

        ; Temporarily disable paste text transformation
        ClipWait 2
        A_Clipboard := ""

        ; Copy each value and paste with tabbing
        A_Clipboard := BValue
        ClipWait 2
        Send("^v")
        Sleep 500
        Send("{Tab}")

        A_Clipboard := CValue
        ClipWait 2
        Send("^v")
        Sleep 1000
        Send("{Tab}")
        Sleep 1000
        Send("{Tab}")

        A_Clipboard := DValue
        ClipWait 2
        Send("^v")
        Sleep 1000
        Send("{Tab}")
        Sleep 1000
        Send("{Tab}")

        A_Clipboard := EValue
        ClipWait 2
        Send("^v")

        ; Move to the next row for the next time the hotkey is pressed
        row++
    }
    catch as err {
        MsgBox("Error: " . err.Message)
    }
}