If you want to be excruciatingly correct, using American English, which would you use?
I would use “racquet” and would call them “racquet sports” but I see a lot of variation here. I guess I wouldn’t look askance at “tennis racket” if I saw it in print–in fact I’ve seen it in print and didn’t want to fix it. But as an editor, I want to fix it.
But not if it’s not broken.
What do y’all think?
I would use “tennis racket” to describe the way a racquet club uses hidden court fees to make extra profit.
I have never seen “tennis racket” referring to the actual implement used to hit the ball. I would assume kayaker’s meaning if I saw it, or maybe “tennis career” depending on the context (as in, “five years ago, I retired from the tennis racket”).
Is this really something out there, like “loose” for “lose”, “your” for “you’re”, and so on? “Your the first player I’ve ever seen loose a game with such an expensive racket.” Gah!
Racquet.
Tennis racket makes me think of game fixing, illegal gambling, protection fees, cement overcoats, etc.
The answer depends on the stylebook. Associated Press Style calls for “racket”:
(From page 248 in the 1987 edition.)
As an American, I would always write “tennis racket,” myself.
For what it’s worth (probably not much), I googled both phrases, and got 5,460,000 results for “tennis racket” vs. 3,600,000 for “tennis racquet.”
Tennis Racket is the way to describe people talking about tennis 
Yeah, but that’s for tenns, not tennis. I always see it spelled “racket” for tenns players. That’s how the great ones, Federr, Samprs, and Lavr, all spelled it, too.
I thought this was a Britishism. Like “organisation” vs. “organization” or “gaol” vs. “jail”. Has “racquet” ever been the standard US spelling?
It’s pronounced “rackay” 
Racquet if you pay for it with a cheque.
Not necessarily. British usage is about half and half. The BBC uses “racquet”; The Times uses “racket”, for example.
Related data point: the sport you normally think of is always “racquetball”, not “racketball”, although apparently there is a thing called racketball, and it’s ironically British, not American: Racquetball - Wikipedia.
Wikipedia calls them “racquet or racket”.
Cambridge online dictionary calls them racket (also racquet).
Dick’s Sporting Goods calls them racquets.
Wison calls them rackets.
Amazon calls them racquets.
Sports Authority calls them racquets
Tennis Express calls them racquets.
Tennis Warehouse calls them racquets.
It looks like most of the folks selling sports equipment spell it racquet.