I am creating Xmas cards with a photo of one of our cats and a caption. Which is correct: Season's Greetings (with apostrophe) or Seasons Greetings (without apostrophe)? Inquiring minds want to know.

Looks like with apostrophe is correct. (Word Choice – Season’s Greetings or Seasons Greetings?)

I found several other sources that are in agreement.

Thanks for that clarification, Artie. I don't want to run afoul of the apostrophe police.

...which I assume is a polite reference to grelber.

Since you're referring (I assume) to a single season, namely the holiday or yuletide season, the correct choice is: Season's Greetings — meaning, of course, 'greetings of the season'.

So let it be written. So let it be done.

Would that be the cats?

Yes, considering that the caption will be added to a photo of Count Kilmousky, one of our cats.

Count Kilmousky mousky kill count?

But as one of the comments above implies, what if you want to be egalitarian and extend a greeting to people of multiple faiths. Then use "Seasons'…".

By the way, to what season does this refer? Winter?

I suspect only indirectly, by implication. The original reference is to the major (religious) holidays in the Judeo-Christian tradition, although winter holidays are quite common across most cultures exposed to associated seasonal hardship.

The season that has been celebrated at this time of year in the Northern hemisphere since time immemorial is THE WINTER SOLSTICE. That time when the sun God (The Great Spirit, Jupiter, Zeus, etc) once again asserted his supremacy over the evil cold of winter and the days would grow longer. The same celebration takes place in the Southern Hemisphere six months later.

The Holy Roman Catholic Church chose the winter solstice as the date to celebrate the birth of Jesus of Nazareth (which is unknown but likely was much later in the spring) to capitalize on the pagan winter solstice celebrations that people all over the world had been celebrating for millennia before the birth of Jesus. Perhaps too it was the church's effort to civilize the frequently bacchanalian fertility celebrations of the time.

As I stated in my previous response, given the question as phrased, it would be the yuletide season; "Season's Greetings" is normally only used in that context. Obviously it could be used for effect in other cases.

For the sake of quasi-completeness: Christmas was previously termed Yule or Yuletide, prior to which was Saturnalia, prior to which were many other solstitial festivals by many other names.

As for the other uses of "season", here's the skinny right from the Mac dictionary:
• a period of the year characterized by a particular climatic feature or marked by a particular activity, event, or festivity: the rainy season | the season for gathering pine needles.
• a fixed time in the year when a particular sport is played: basketball season is over.
• the time of year when a particular fruit, vegetable, or other food is plentiful and in good condition: the pies are made with fruit that is in season | lobster season.
• an indefinite or unspecified period of time; a while: this most beautiful soul, who walked with me for a season in this world.
• archaic a proper or suitable time: to everything there is a season.

True that JS originally stated it was a Christmas card, and as joemikeb explained, the season was co-opted by the Catholic Church; but make no mistake—anyone who is not Christian or pagan would not see "yuletide" or "season's greetings" as relating to their religious beliefs.

And don't get me started on the (misguided, in my opinion) belief that Christmas trees are not religious symbols!