I believe that the key inference is:"If what is required in any legislation is an act of reciprocity and recognition, what is involved in any such recognition is a recognition of an object". Whether or not this is true (and I suspect that most pragmatists would see it as a non-sequitur, if intelligible at all) this is certainly not Kant's position. In the Kantian scheme "object" is a category of appearances, while the moral autonomy belongs to the self beyond the appearances. The ultimate dictates of practical reason therefore can not rely on recognition of objects, they are synthetic a priori, like the categorical imperative. But even in other forms of transcendental idealism, like Husserl's, objects and the rest of "objective reality" are constituted as such by the "transcendental subject", and can not therefore be involved in a fundamental philosophical account of their constitution. Pragmatists and transcendental idealists agree on that much.
In fact, Kant's theory of autonomy and free will goes far beyond what any pragmatist would countenance. Not only is self's will objectless, it is atemporal as well, beyond the realm of phenomena and their causal chains, affecting them only as a totality, see Is Kant's "noumenal self" argument on freedom flawed? Hence the demand of the categorical imperative to adopt universalizable maxims. There is a big difference between Kant and Hegel on the source of this autonomy however. To Kant the "transcendental subject" is some pre-empirical commonality residing in the "supersensible substrate of humanity", and it is exactly on this point that Hegel broke with him, interpreting it instead as the product of social Geist produced by sedimentation of historical developments. Brandom goes further in demythologizing this picture by identifying the Geist with historical human communities and their practices and customs.
Pippin discusses Kantian (and Hegelian) self-legislation and autonomy in his book Persistence of Subjectivity, in the chapter on Brandom's (much more Kantian) fellow pragmatist, McDowell:
"There is, for Kant, only one way of realizing this requirement (whatever it is) of self-direction or self-determination:
Following the dictates of practical reason, to do nothing that you cannot rationally justify. To be autonomous is to constrain our action only by rules and principles we can justify with reason, that we can defend to ourselves with reason. (We can thereby “stand behind” our actions, as our own.) Finally, though, there is for Kant only one way to realize this condition, since pure practical reason, according to him, has no insight into which ends are universally worth pursuing... Yes, in means-end reasoning, we should deliberate about the most efficient means, and perhaps we can say we should have a life plan, a way of ordering and prioritizing our ends. But that has nothing to do with autonomy, since it is not ultimately up to us which overall ends, even very general ones, appeal to us. Likewise, this prudential interpretation is too weak to rope in Kant’s unique position...
“Moral insight” is of a peculiar sort in Kant; seeing that anything is the case – say, that this is the principle that a purely rational being with a finite will would adopt – in itself in no way constrains our action, does not function just thereby as a norm. He always uses language like “being obligated is placing ourselves under the law,” submitting or subjecting ourselves (unterwerfen) to it. So if we consider the question of Why are we obligated to act on maxims consistent with the demands of pure practical reason? we have to say something like “Because we have obligated ourselves to it.” The self-legislation language – I make this a law for myself, submit to its authority – is not expungeable from Kant..."