The colloquial phrase can eat an apple through a picket fence, and its variants, are used of a buck-toothed person.
—Cf. the phrase couldn’t stop a pig (in a gate), which similarly derides a person’s physical characteristic.

These are, in chronological order, the earliest occurrences of the phrase can eat an apple through a picket fence, and variants, that I have found:

1-: From the column Main Street Meditations, by Eleanor Clarage, published in the Cleveland Plain Dealer (Cleveland, Ohio) of Saturday 3^rd June 1933:

You Get the Idea.
Describing a girl with buck teeth, a gent at last night’s party had this to say: “She could eat an apple through a picket fence without cutting her lips.”

2-: From the Detroit Times (Detroit, Michigan) of Saturday 28^th October 1939—Leo Fitzpatrick, guest columnist in the absence of Leo Macdonell, sports editor of the Detroit Times, wrote the following to the latter about Dick Richards, “anchor man of the Detroit Lions”:

One night Dick was on his way to see the Red Wings compete for the Stanley Cup (I forgot to tell you he’s a rabid hockey fan). As he was leaving a downtown hotel the doorman, whom he had known for some time, was eating an apple. Did you ever try to eat an apple through a picket fence, Leo? Well, that’s what it looked like. Dick immediately became sympathetic. Told him to go over to a certain dentist. Two weeks later a bill for $80. But the next time I saw the fellow he could eat caramels without having them stick to the upper plate.

3-: From Overbite? Chew Bad, Says Navy, published in The San Antonio Light (San Antonio, Texas) of Sunday 28^th April 1940:

Do your teeth encircle the cob when eating spring corn?
Can you eat an apple through a picket fence?
If your answers to these questions are in the affirmative, you’ll never be a son of the modern navy.
With the innovation of new fighting equipment for the United States navy, physical requirements for enlistment have become more stringent than ever, according to L. E. Park, head of the naval recruiting station here.
And one of the new requirements is that your teeth do not have an overbite over one-sixteenth of an inch.
The reason is that if your top teeth stick out too far, you won’t be of much use to the navy in case you are stationed to a submarine division.
Should a submarine be trapped at the bottom of the ocean, as happened recently in the case of the Squalus ¹, Park explains, sailors who wish to escape must be able to hold a Momsen lung ² in their teeth.
In the latest submarines, hatches are provided which enable the crews of trapped boats to escape through openings in the sides and float to the surface of the water.
The Momsen lung provides oxygen to the sailor during the escape but must be gripped tightly in the mouth to prevent water seepage. Hence the new teeth regulation.

¹ On Tuesday 23^rd May 1939, the Squalus, a U.S. submarine, sank off the coast of New Hampshire. The sinking drowned 26 crew members, but an ensuing rescue operation, directed by Charles Momsen ², saved the lives of the remaining 33 aboard.
² Invented by the U.S. naval officer Charles Momsen (1896-1967), the Momsen lung was an underwater rebreather used before and during the Second World War by U.S. submariners as an emergency escape device.

4-: From Editorial, published in The Nance County Journal (Fullerton, Nebraska) of Thursday 12^th September 1940:

The army won’t enlist men with pronounced buck teeth. We don’t know what the test is, but what about seeing if the candidate can eat an apple through a picket fence?

5-: From the column New York Cavalcade, by Louis Sobol, published in The Evening News (Harrisburg, Pennsylvania) of Saturday 22^nd March 1941:

S. A. Scheier, sponsor of the Gag File (Big Laughs and Little Laughs, column lengths and fillers), still insists I need his service. . . . He sends me a few samples, free of charge: “She has such buck teeth, she could eat an apple through a picket fence” . . . “The bags under his eyes are so large, his nose looks like a red cap.” . . . “Is that your face, or another undeclared war?”

6-: From the column Getting Around, by Ted Ashby, published in the Des Moines Tribune (Des Moines, Iowa) of Monday 13^th April 1942:

OVERHEARD IN DOWNTOWN TAVERN: “WITH THOSE TEETH, HE COULD EAT AN APPLE THROUGH A PICKET FENCE.”

7-: From the column Talking it Over, published in The Brewton Standard (Brewton, Alabama) of Thursday 11^th June 1942:

I see where Dorothy Thompson ³ and Eleanor (I Can Eat An Apple Through a Picket Fence) Roosevelt ⁴ are still holding forth in the columns of some daily papers. For pure unadulterated ego, those gals take the prize.

³ Dorothy Thompson (1893-1961) was a U.S. journalist.
⁴ Eleanor Roosevelt (1884-1962) was a U.S. humanitarian and diplomat. She married the U.S. Democratic statesman Franklin Delano Roosevelt (1882-1945) in 1905.

The earliest occurrences of the variant can eat an apple through a tennis racquet that I have found are:

1-: From Ned Cronin’s sports column, published in the Daily News (Los Angeles, California) of Friday 7^th July 1950:

Up until the time when my dentist crawled into my mouth with a brace and bit and a balpeen hammer, I was one of the few people in the country who could eat an apple through a tennis racquet. I am now able to eat a tennis racquet through an apple. And with practically no upper plate rattle at all.

2-: From Farmers Market Today, by Dick Kidson, published in the Los Angeles Times (Los Angeles, California) of Wednesday 23^rd February 1955—the following is about the Ken-Mar Optical Shop:

Frames are now designed to draw attention to one’s best features and to minimize any facial deficiencies. Now you take a gal who is a little unfortunate. Her teeth may be such that she could eat an apple through a tennis racquet. Yet, designs like a sunburst on the frames of her glasses will divert the eye to the upper portion of her face, which may be lovely.

The following is an occurrence in Australian English—from The Western Herald (Bourke, New South Wales) of Friday 23^rd October 1959:

Chad Morgan, the only man who has eaten an apple through a tennis racquet and not broken any strings. See him personally. Bourke Memorial Hall, Friday 23 and Saturday 24 October.

Finally, the earliest occurrence of the variant can eat an apple through a letter box that I have found is from the Daily Mirror (London, England) of Saturday 17^th November 1979:

Coventry skipper Tommy Hutchison has been ribbing Jim Holton over his two new front teeth. Cracks Hutch: “I think the dentist got them mixed up with those for Bugs Bunny. They protrude so much he could eat an apple through a letter box.” Perhaps the new chant will be: “Six-foot-two, eyes of blue, big Jim’s Fangs are after you . . . .”