A quick google search has led me to conclude that the word appeared in the late 15th century--which would have been plenty of time for Shakespeare himself to learn it, I think.
I'm curious how widely known the word might have been. Would nobles going to Shakespeare's play recognize the use of metaphor and be able to name it? What about merchants and peasants? (I'm sure all these people were capable of understanding metaphor--I wonder if they knew that particular term).
EDIT: Spelling
The OED gives the following uses of the word before Shakespeare's time:
a1500 (▸c1477) T. Norton Ordinal of Alchemy (BL Add.) (1975) 63 (MED), Thei made theire bokis to many men ful derk, In poyses, parabols, & in methaphoris alle-so, which to scolers causith peyne and wo.
1533 King Henry VIII in Wotton Lett. (1654) Suppl. 8 And rather then men would note a lye when they know what is meant, they will sooner by allegory or metaphor draw the word to the truth.
1555 E. Bonner Certaine Homelyes 71* Chryste alwayes in hys speakynge dyd vse fygures, metaphores and tropes.
1563 W. Baldwin et al. Myrrour for Magistrates (new ed.) Collingbourne xxxvii, These metafors I vse with other more.
1600 S. Nicholson Acolastus his After-witte 861 Mens words are Metaphors, it makes no matter.
Furthermore, Shakespeare uses the word himself four times within his plays (thrice in All's Well that Ends Well and once in Twelfth Night). See them here at the Open Source Shakespeare Concordance.
Words with similar meanings, such as allegory or simile, had existed even earlier in the English language than metaphor (which was around in Old and Middle French for centuries). It seems to me that yes, Shakespeare's audience would've understood what the word metaphor meant.