Is there any way my dad can get VA benefits if I can't find his discharge papers?

Also, if a Merchant Marine ship was taken under command by the USN,directly, those Merchant Marines came under the direct authority of that ship's captain...which I assume, was a USN officer. That should qualify as service time in the USN. See section 39.
http://www.benefits.va.gov/warms/docs/admin21/m21_1/mr/part3/subptiii/ch02/ch02_secf.doc

If your father has a VA card, they should have already documented his eligible veteran's status at the VA. But leave it to the VA to continually require that documentation to be presented again and again

In WWII they did not issue a DD-214. Look for a "Military Record and Report of Separation" also known as "WD AGO form 53-98"

I just googled how to get military records for my deceased father so that my mother could get benefits. I found this site: https://www.archives.gov/research/military. I filled out an online form, printed out and had mom sign a document, scanned it in and within two weeks had a copy of Dad's DD-214 in hand.