One thing to keep in mind is that Bourbaki started in the 1930s, so in some sense simply *too early* to include category theory right from the start on, and foundational matters where rather fixed early on and then basically stayed like this. Since (I think) the aim was/is a *coherent* presentation (as opposed to merely a collection of several books in similar spirit) to change something like this 'at the root' should be a major issue.
Some 'add on' seems possible but just does not (yet) exist; and it seems the idea to write something like this was (perhaps is?) entertained (see below).

To support the above here is a quote from MacLane (taken from the French Wikipedia page on Bourbaki which contains a somewhat longer quote and source):

Categorical ideas might well have fitted in with the general program of Nicolas Bourbaki [...]. However, his first volume on the notion of mathematical structure was prepared in 1939 before the advent of categories. It chanced to use instead an elaborate notion of an échelle de structure which has proved too complex to be useful. Apparently as a result, Bourbaki never took to category theory. At one time, in 1954, I was invited to attend one of the private meetings of Bourbaki, perhaps in the expectation that I might advocate such matters. However, my facility in the French language was not sufficient to categorize Bourbaki.

There it is also mentioned that (in the context of the influence of the lack of categories on the discussion of homological algebra, only for modules not for abelian categories):

On peut lire dans une note de bas de page du livre d'Algèbre Commutative: « Voir la partie de ce Traité consacrée aux catégories, et, plus particulièrement, aux catégories abéliennes (en préparation) », mais les propos de MacLane qui précèdent laissent penser que ce livre « en préparation » ne sera jamais publié.

This translates to (my rough translation): One can read in a footnote of the book Commutative Algebra: "See the part of this Treatise dedicated to categories, and, more specificially, to abeliens categories (in preparation)", but the sentiments of Mac Lane expressed above [part of which I reproduced] let one think that this book "in preparation" will never be published.

The precise reference for the footnote according to Wikipedia is N. Bourbaki, Algèbre Commutative, chapitres 1 à 4, Springer, 2006, chap. I, p. 55.

Twenty-five years with Nicolas Bourbakiams.org/notices/199803/borel.pdf on page 378, where a short account of the story of thecongrès du foncteur inflexibleis given. It discusses Grothendieck's proposal how they should treat sheaf theory and why that route wasn't chosen. $\endgroup$ – Martin May 24 '13 at 0:23