Try array-entering this in I2 and filling down.
Formula:
=IF(SUM((INDEX('Days Off'!$A$3:$Q$30,N(IF(1,ROW($1:$28))),N(IF(1,MATCH(D2:G2,'Days Off'!$A$2:$Q$2,0))))>=B2)*(INDEX('Days Off'!$A$3:$Q$30,N(IF(1,ROW($1:$28))),N(IF(1,MATCH(D2:G2,'Days Off'!$A$2:$Q$2,0))))<=C2)),"Yes","No")
If you aren’t familiar with array-entered formulas array enter means the formula must be committed from edit mode by simultaneously pressing and holding down Ctrl and Shift while hitting Enter.
PS I changed some dates to test this a little.
Bookmarks