Searching the literature in which the existential quantifier first appears (from Peano in the 1890’s up to the middle of the twentieth century) shows a fairly clear picture:

• Peano (from the Formulario, 1899-1908) used his “serif” ∃, but for a set–former, not for an existential quantifier, and he never put a bar through it.

• Whitehead & Russell, Principia Mathematica (1910–13) are the first to use Peano’s sign as an existential quantifier. When they want to deny existence they simply prefix the ordinary negation sign: ~∃x φ(x). No barred ∃ appears.

• Hilbert & Ackermann, Grundzüge der theoretischen Logik (1st ed. 1928, 2nd ed. 1938), Hilbert & Bernays, Grundlagen der Mathematik (1934/39), Tarski’s papers of the 1930s, the books of Quine (1934, 1940), Church (1944) etc. all follow the same practice: ¬∃ or ~∃, never ∄.

The first place where one actually finds the slashed existential sign in print is in the internal cyclostyled fascicles of

    N. Bourbaki, Éléments de mathématique. Livre I : Théorie des ensembles
    (Fascicule de résultats, 1939; first public printing 1954).

Already on p. 6 of the 1939 stencil (p. I-6 of the 1954 edition) one reads, for example,

    ∄ x (x ∈ ∅) ,

and the notation is listed without comment among the “Convention de signes” at the beginning of the work. Bourbaki are simply applying Peano’s older idea of “negative made with a bar through the positive symbol” (Peano had given ∉, ∌, ⊄, …) to the newer quantifier ∃.

After Bourbaki the barred sign spread quickly. It is used, for instance, in:

• Kemeny, Snell & Thompson, Introduction to Finite Mathematics (1957)
• Addison, Mathematical Logic (1961 lecture notes)
• ISO 31-11 “Mathematical signs and symbols” (first draft 1968)
and from the 1970s it is available in every standard mathematical font.

Therefore, as far as published sources are concerned, the earliest documented use of the symbol ∄ for “there does not exist” is due to the Bourbaki group in their 1939 (published 1954) Théorie des ensembles.