Matching dates and returning another column

This seems to do it. I introduced the use of Indirect into the formula so you can just enter the formula in E3 (confirmed with Ctrl + Shift + Enter) and then drag over to F3 and then drag down.

The Indirect part of the formula looks at E2 and F2 for the sheet name which makes the formula dynamic. The one draw back, the Indirect function is volatile so you want to use it sparingly in your spreadsheet.

If you happen to have 2007+ you can invoke the IFERROR in lieu of ISNA which would reduce the formula to...

CSE
=IFERROR(INDEX(INDIRECT("'" & E$2 & "'!$G$4:$G$138"),MATCH(1,IF(INDIRECT("'" & E$2 & "'!$C$4:$C$138")=$B3,IF(INDIRECT("'" & E$2 & "'!$D$4:$D$138")=$C3,1))),0),"")

Could be a smarter way, but this is the route I would go...