When did Grothendieck join Bourbaki? Bourbaki listed Grothendieck as a third-generation member. Nevertheless, it does not provide details on when he joined and when he left.
Concerning his departure, there is a
Letter from October 9, 1960 and a translation¹

I thank You for your letter², marked by both wisdom and clemency. Indeed
it seems pointless that a personal disagreement could be the occasion for the
departure of a disciple. I recognize that it was pointless for me to wait for
the Master to arbitrate a quarrel that did not concern him and that such
arbitration would resolve nothing.
I have asked myself many times over the years of my collaboration with
the Master whether my lack of social skill, my impassioned character, and
my repugnance for overcoming the repugnance of others, did not render me
unsuitable for a productive collaboration during the meetings. No longer
wanting to search for the cause anywhere except in myself, I now think that
it is better this way and that I reached earlier than the traditional age the
moment when I would better serve the Master by my departure, rather than
remaining as a result of His kind insistence.
I will endeavor to remain worthy of the teachings that You for so long
lavished upon me and not to betray the spirit of the Master who, I hope, will
remain visible in my work as it has been in the past.

With this, we can infer the date he left Bourbaki.
Regarding his entrance, It is very likely that he entered with P Cartier, but I cannot prove that. According to Bourbaki that was in 1955.


Is there a reference about when Grothendieck entered Bourbaki?




*

*By W Messing

*It would be very interesting to know this letter too.

 A: There is evidence that Grothendieck was a member of Bourbaki in 1956, from "La Tribu”, internal reports on the activities of the group. In the course of a discussion on the tentative contents of Theory of Sets it is written about chapter V that:

Chap. V (Catégories et foncteurs) - Pour commencer Grothendieck
rédigera une espèce de Fascicule de Résultats en style naïf, afin que
Bourbaki se rende compte de ce qu'il est utile de pouvoir faire. On
formalisera ensuite. (La Tribu, 39, 4 June-7 July 1956).

 source, page 337.
To begin with, Grothendieck will write a sort of Results Booklet in a naïve style, so that Bourbaki will realize what can usefully be done. We will then formalize.
Volume V never appeared, this MO thread discusses the conflict which might have been the reason for Grothendieck to leave Bourbaki.
A: I propose February 3, 1950 as a possible answer. The following appears at the bottom of page 3 here:

Grothendieck was one of five “cobayes” (guinea-pigs; young mathematicians invited
to see whether they would be suitable for Bourbaki and Bourbaki for them) at a Bourbaki meeting which took place in Nancy, Feb. 3-7, 1950. The other “cobayes” were F.
Bruhat, Braconnier, Berger and Riss; the actual members were Cartan, Chabauty, Delsarte, Dieudonné, Ehresmann, Godement, Mackey, Pisot, Roger, Samuel, Serre, Schwartz.

The link above coming from the in biography titled Who Is Alexandre Grothendieck? Anarchy, Mathematics, Spirituality, Solitude by Leila Schneps where it is in Chapter 3: From student to celebrity (1950-1952).
A: In Notices of the AMS Volume 63, Number 3 we find in the section of P Cartier:

It was in March 1955 at the Bourbaki seminar after a special lecture that
Grothendieck gave about convexity inequalities¹. He told me: “Very soon, both of us will join Bourbaki.” I began regularly attending Bourbaki meetings in June 1955. Grothendieck joined soon and participated actively from 1956 to 1960. In June 1955 one of the most interesting pieces to read during our meeting was a first draft of his famous Tôhoku paper, where he gives a new birth to homological algebra.

And from Grothendiec-Serre correspondence
Grothendieck to Serre June 4,

You will find enclosed a neat draft of the outcome of my initial reflections on the foundations of homological algebra.

Serre to Grothendieck on July 13,

Your paper on homological Algebra² was read carefully³ and converted everyone (even Dieudonné, who seems to be completely functorised) to your point of view.



*

*Réarrangements de fonctions et inégalités de convexité dans les algèbres de von Neumann munies d'une trace.
Séminaire Bourbaki : années 1954/55 - 1955/56, exposés 101-136, Séminaire Bourbaki, no. 3 (1956), Exposé no. 113, 13 p.


*Some aspects of homological algebra


*At the Bourbaki meeting
A: A very good interview in French with former Bourbaki members Jean-Pierre Serre, Pierre Cartier and Jacques Dixmier, where the date is given, at the mark of 17:40.
[Dixmier]: So we have spoken about how we became members [of Bourbaki], let's talk about others. Grothendieck was enlisted as a member of Bourbaki when he was in Nancy.
[Serre]: No, he was not a member of Bourbaki when he was in     Nancy, he was a student of a member, that is not the same thing.
[Dixmier]: He was a student of Schwartz and Dieudonné, they said immediately: He is amazing.
[Serre]: Still, this is a very different thing, one did not consider taking him on as a member right away.
[Dixmier]: Well, I would say that he was recruited in 1955, after you [Cartier].
[Serre]: I would say that as well.
[Cartier]: Around the same time.
