Claire-
After reviewing your scenario a little bit further I think these two formulas will work for what you need. You will need to adjust the bold portion of the formulas to only encompass the month range (i.e. July) you are looking for:
July Example (Put this in cell C3 and drag down):
Formula Based on Employee Name Criteria:
=SUMPRODUCT(('[Semester 2 Formula Help.xlsx]Class Support - South Region'!$B$5:$B$20=LOOKUP("Z",A$3:A3))*('[Semester 2 Formula Help.xlsx]Class Support - South Region'!$E$5:$M$20=B3),'[Semester 2 Formula Help.xlsx]Class Support - South Region'!$E$6:$M$21)
Formula Based on Employee ID Criteria:
=SUMPRODUCT(('[Semester 2 Formula Help.xlsx]Class Support - South Region'!$B$6:$B$21=SMALL(A$4:A4,1))*('[Semester 2 Formula Help.xlsx]Class Support - South Region'!$E$5:$M$20=B3),'[Semester 2 Formula Help.xlsx]Class Support - South Region'!$E$6:$M$21)
Hope this helps.