Note: the values in E5:E7 are actual dates, formatted with the custom number format "mmyy".
Working from the inside out, the expression:
uses the TEXT function to generate an array of strings in the format "mmyy":
which are compared a single string based on the value in E5, "0117". The result is an array of TRUE / FALSE values:
which are fed into the MATCH function as the lookup array, with a lookup value of TRUE, and a match type of zero for exact match. In exact match mode, the MATCH function returns the position of the first TRUE in the array, which is 1 in the formula in F5. This position goes into INDEX as the row number, with an array based on the named range "entry":
Index returns the item inside entry at the specified position.
Note: if an entry isn't found for a given month and year, this formula will return #N/A.
First entry based on today's date
To get the first entry for a given month and year based on today's date, you can adapt the formula to use the TODAY function instead of the value in E5: