It's extremely interesting that an early mention of tarot comes from John Florio. I didn't know that about Florio, but in relation to the Shakespeare writings, the name of Florio is about the best a person could want. If a person had his choice, of all the names in England, you'd have trouble finding anybody better than Florio, in re tarot and Shakespeare.
It's possible to bring Florio and Shakespeare very close, although there's no surviving documentation of their actual acquaintance, as far as I know. Florio was a tutor to the Earl of Southampton, the aristocrat to whom Shakespeare's Venus and Adonis, and Rape of Lucrece are dedicated. It puts Florio and Shakespeare only one person apart, through Southampton.
John Florio dedicated his Italian-English dictionary to Southampton, the “most Honorable Earl of Southampton, in whose pay and patronage I have lived some years.” Florio's employment for Southampton apparently began around 1591 or so, and the Shakespeare V & A dedication to Southampton was published in 1593.
Also, the title of Love's Labours Lost is generally accepted to have come from a line in either Florio's "First Fruits," or "Second Fruits" (I've forgotten which.) The quote being: "We need not speak so much of love, all books are full of love, with so many authors, that it were labor lost to speak of Love." It seems clear enough that Shakespeare was familiar with Florio's books, to take a play title from them. There are various other indications in the plays that the Bard knew Florio's writings. As another example, it's now taken for granted that Montaigne was a source for The Tempest, and Florio (among others) translated Montaigne.
The year 1598, for Florio's mention of tarot, is excellent in relation to the composition of Hamlet, since the First Quarto of the play was printed in 1603. That puts some 5 years between Florio's printed mention of tarot, and the apparent allusions to tarot by Shakespeare in Hamlet, in print, which is plenty of time.
The Antony and Cleopatra quote that Fulgour posted is good for timing, too. The first documented evidence of the existence of A & C is a Stationers' Register entry in 1608, which is ten years after the Florio mention of tarot. That gives a full decade for the Bard to be familiar enough with tarot to allude to it in A & C, based simply on Florio, and the known documentary record.
The King John remark about "cards" that I posted is closer for timing. The earliest document mentioning King John is a list of plays by Francis Meres in 1598. Same year as the Florio dictionary. Still, hardly impossible.
I'm sure there are other Shakespeare writings - as Ross points out - in which playing cards are mentioned, or implied, and some of them will likely pose a problem in relation to Florio's published statement defining tarot. There'll be cases where the play is earlier than 1598. I see no problem in that, since Florio must certainly have known of tarot long before he defined it in print, and it's possible to place Florio somewhere close to Shakespeare, through Southampton, as early as 1593. (And even failing that, the 1592 publication by Delamothe covers it.)
So it appears to me the timing of it all works nicely, for Shakespeare to allude to tarot in his writings. And as to a source, you'd hardly do better than Florio.
Thank you very much, Ross! There's no absolute proof, but the more I learn, the better it looks, for Shakespeare knowing tarot.