Have one column automatically populate with data from another column that for a date range

[fireking14]

On the attached document, I am trying to have the columns: Part #, Description and Cost of the table in sheet "March" automatically populate with the data from the corresponding columns in the sheet "Running Total", but only the ones that fall into the date range specified in the sheet "March." I need to do this without a macro and if possible no VBA formulas. I will be using a macro later on to remove the duplicates that will exist as more and more data is entered into the running total sheet.

[FlameRetired]

Withdrawn by FR. Formula not returning desired results.

[FDibbins]

Try this ARRAY formula in A2...
=IFERROR(INDEX('Running Total'!A:A,SMALL(IF(('Running Total'!$E$2:$E$7>=$I$1)*('Running Total'!$E$2:$E$7<=$J$1),ROW('Running Total'!$A$2:$A$7)),ROWS($A$1:A1))),"")
...confirmed by pressing CTRL+SHIFT+ENTER to activate the array, not just ENTER. You will know the array is active when you see curly braces { } appear around your formula. If you do not CTRL+SHIFT+ENTER you will get an error or a clearly incorrect answer.
Then copy down and across as needed