Here's a basic timeline of the word, grub, based on the earliest uses of it, as outlined by The Oxford English Dictionary...
Verb:
1300...to dig by mankind, as in someone farming, and from Old Teutonic
1647...to dig, as by animals searching for roots to eat
1725...to eat.
Noun:
1400...a dwarf
1420...an insect larva
1653...a literary hack
1659...food.
So, yes, it dates back to the 1300s, but it did not mean food (noun) for a further three-plus centuries. Of course, words may have been in use in SPEECH before they ever appeared in writing, but the fact remains that the only evidence for them is the evidence we possess in WRITING. Accordingly, I stick with my earlier responses above.