This is the "form" of the formula to get what you want:
Formula:
=COUNTIFS('Raw Data'!$J$2:$J$328,">="&H8,'Raw Data'!$J$2:$J$328,"<="&I8)
Your data has problems.
The dates in Raw Data $J$2:$J$328 and Raw Data $O$2:$O$328 are all text and have to be converted to real dates. You can do this by inserting helper columns next to columns J and O.
In the new column K enter this formula and copy down the length of the data:
Formula:
=DATE(YEAR(J2),MONTH(J2),DAY(J2))
Format as yyyy/mm/dd
Copy column K and Paste Values into column J and then delete the helper column. Repeat this for column P and copy and paste values back into column O.
The other problem that you have is that the dates have no data for the dates that you have specified. Start your dates earlier in the year and you won't run into errors until the dates don't have data. As it is, you can't check the formulae for correctness because they will always produce an error.
To eliminate the errors in your final formulae enter the formulae like this:
Formula:
=IFERROR(COUNTIFS('Raw Data'!$J$2:$J$328,">="&H8,'Raw Data'!$J$2:$J$328,"<="&I8),"")
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