History

Exactly where the Vodkatini fits in the vexed question of the origins of the Dry Martini are unknown but it's certain that 'Vodka Martinis' were being served in New York after the repeal of Prohibition in 1933. Its first known print appearance is in a 1935 Smirnoff marketing pamphlet:

VODKA MANHATTAN COCKTAIL
2 parts Vodka
1 part Italian Vermouth (Sweet)
Dash of Angostura Bitters
Ice and stir well. Strain into cocktail glass. Add Maraschino Cherry.
VODKA MARTINI COCKTAIL (DRY)
3 parts Vodka
1 part French Vermouth (Dry)
Put 1 or 2 pieces of lemon peel with ice in mixing glass, stir well. Strain into cocktail glass.
Twist a bit of lemon peel on finished drink. Add an olive if desired.

Smirnoff brochure, 1935

The Vodka Martini's first appearance in a cocktail recipe book is in Lucius Beebe's 1946 The Stork Club Bar Book, which has two recipes, one in the "Night" section and the other in the Appendix.

Most personal preferences in mixed drinks are simply variations on standard or basic themes in much the same manner that all witticisms are supposedly based on seven fundamental joke structures. Carole Landis, for example, swears by a vodka Martini which, if pursued over any length of time, she promises will make the customer scream next morning for a Bloody Mary.

Vodka Martini #1:
3 oz. vodka
1 oz. dry vermouth
Stir or spoon with cube of ice and serve as a conventional Martini in 4 oz. glass.

According to the sworn testimony of Joe Acre, captain, the vodka Martini is a favorite with Dashiel Hammett:

Vodka Martini #2:
⅓ oz. French vermouth
⅔ oz. vodka
Serve in a cocktail glass. Serve very cold.

Lucius Beebe, The Stork Club Bar Book, 1946

This is followed six years later by Ted Saucier's 1952 Bottoms Up.

VODKATINI
By Jerome Zerbe, Photographer, and Society Editor of Town and Country
4/5 jigger Smirnoff vodka
1/5 jigger dry vermouth
Ice
Stir in mixer. Strain into cocktail glass, and add twist lemon peel.

Ted Saucier, Bottoms Up, 1952

Saucier credits the invention of the Vodka Martini to Jerome Zerbe (1904-1988), the society editor of Town and Country Magazine. Zerbe is also credited for being the first celebrity paparazzi. A socialite, he was almost as much a celebrity as the celebrities he photographed and counted movie stars such as Gary Cooper, Errol Flynn, and Cary Grant as close friends.

The popularity of the Vodkatini was helped by the rise of vodka during the 1980s and 1990s, and Ian Fleming's character James Bond, who famously drinks Vodka Martinis "shaken, not stirred." [Bond actually drinks Vesper Martinis.]