In some respects, you could say that Sartre is "borrowing from Kant." It will greatly depend on what you mean by borrowing. Iphigenie's comments are highlighting the differences, and those are definitely worth pointing out but a type of "rationalist" heritage is worth bringing up.
What I would say is the common thread is an emphasis on "autonomy" and a belief that human beings possess a type of unrestricted autonomy in reference to their choices. Neither thinker is ultimately naive enough to say that we always have this, but both theories have been accused of separating us from bodily concerns under the belief that we can make choices regardless of circumstances. In the simplest terms, both have a very high belief in our autonomy and that we have a free rational will.
At the same time, there are some crucial differences about how our rationality works for both thinkers. A core feature of Kant's account of morality and his account of rationality is that they have universality. In moral theory, this means that the moral rules apply to everyone insofar as they are humans at all. (Kant's particular method of how he gets there for morality is both highly contested and largely believed to have failed). But the key is that for Kant, reason works the same in every person and will logically lead them to the same conclusions. This is evident if you consider both the phenomenological / noumenal split and the categories of the understanding (the transcendental unity of apperception). This has led to the accusation that Kant's philosophy is a "German sausage machine" because whatever comes in gets pumped out the same. Thus, the CI is the same for all subjects who are rational. In a sense rationality predominates autonomy. (There's a lot more complexity if we are start looking at imperfect duties...)
For Sartre, our autonomy is our defining feature. In his view, we do seem to have a type of practical rationality, but the questions of what we will make our choices based are on largely left up to our autonomy to decide. Thus, in Existentialism is a Humanism, Sartre works from the case of the young man who must decide whether to serve France, his mother, or the church. For Sartre, we will never have enough evidence to decide the most important questions rationally (i.e., there's no sausage machine -- there's rats, pigs, whales, and chickens and you've got to decide what they are there for).
To sum up, both share a belief in autonomy and some idea of rationality, but for Sartre both are ultimately species of autonomy. For Kant, autonomy is about a will ordered under reason. In a sense, they are both inheritors of Descartes and the outcome of the Meditations where we have faculties of will and reason -- and one can outrun the other and go wrong. For Descartes and Kant, will is the source of error. For Sartre, false belief in "reason" is.