no, of course Bourbaki was not the first to notice the need. Probably the first relevant instance was in Dedekind, who was of the opinion that "nothing is more dangerous in mathematics, than to accept existences without sufficient proof" (letter to Lipschitz, 1876; see also his famous letter to Keferstein, 1890). He realized the need for a proposition securing the existence of an infinite set (proposition no. 66 in his booklet -- not no. 6, is this a mistake in Suppes?), the big difference with us is that he thought he could prove it by purely logical means. Dedekind seems to have been motivated by Bolzano's "Paradoxien des Unendlichen" (1851) in giving his proof, and probably also in realizing the need for it.
Hilbert, Russell, and many others seem to have accepted Dedekind's proof. But after the publication of the Russell paradox in 1903, widely publicized in the books of Russell and Frege, criticism of Dedekind began to appear. One of the first instances is in Hilbert's paper 'On the foundations of logic and arithmetic', published in 1905 (included in van Heijenoort, "From Frege to Gödel", 1967). Subsequently, the proposition was transformed into an axiom by Hilbert's colleague Ernst Zermelo. In Zermelo's axiomatization paper of 1908 (also in van Heijenoort), he calls it "Dedekind's axiom" as a way of acknowledging the importance of that precedent.
The axiom systems for set theory became popular during the 1920s and 1930s. Also within other systems usual at the time (type theory), people adopted explicitly an axiom of infinity -- see e.g. the famous papers on mathematical logic by Gödel (1931) and Tarski (1933 and 1935). There were also interesting philosophical discussions about this topic, e.g. by Ramsey. By the time of Bourbaki's presentation, it was only too well known that one needed this axiom.
Best wishes to all, and peace on earth,
Several weeks ago Alexander Zenkin (email@example.com), in a post to HM which unfortunately I did not keep, said that Bourbaki stated the need for Axiom of Infinty: There exists an infinite set
Was this argument first made by Bourbaki?
In Patrick Suppes _Axiomatic Set Theory_ 2nd (?) edition New York: Dover The attempt to prove the existence of an infinite set of objects has a rather bizarre and sometimes tortured history. Proposition No. 6 of Dedekind's famous Was sind und was sollen die Zahlen?, first published in 1888, asserts that there is an infinite system. (Dedekind's systems correspond to our sets.) [footnote] A similar argument is to be found in Bolzano [Paradoxien des Unendlichen, Liepsiz 1851] section 13 <end quote>