The answer to this depends on how you define what a right is. The pursuit of self-actualization is not a legal right that is protected by the Constitution of the United States. However, you could argue that it is a human right that people have even if it is not explicitly granted to them by any government.
It is possible to argue that a right is something that is guaranteed by law. In other words, we in the United States have the right to freedom of speech because the First Amendment clearly says that we do, but we do not have the right to wear whatever kind of clothing we want because there is no specific right to do so that is protected by law. By this definition, there is no right to pursue self-actualization. The Constitution sort of implies that we have this right since we are able to say what we want, to read what we want (freedom of the press), and to worship however we want. But it does not say that we have a right to pursue self-actualization.
It is also possible to argue, however, that there are rights that exist even if no government recognizes them. If such rights do exist, the right to pursue self-actualization could certainly be one of them. As human beings, we are constantly trying to improve ourselves. It seems clear that we have an impulse in this direction. Therefore, self-actualization is a basic human function much like the desire to procreate or the desire to be loved. If this is true, then the pursuit of self-actualization would be a human right.
From this, we can see that the pursuit of self-actualization is not a formal right granted by the government, but it may be a basic human right.
Whether self-actualization is a right is a matter of opinion, but then again, just about every right is a matter of opinion. The Declaration of Independence states that Americans have the right to life, liberty, and the pursuit of happiness, but the execution of this can't be fully enforced in any way and can be rejected by any individual with varying consequences. It's really hard to say what is considered a right or not, and this could be thought of as shaped by the norms and standards of society.
Self-actualization generally means reaching one's potential and being able to grow. Abraham Maslow listed this as the top of his hierarchy of needs, but does it being a need necessarily mean it's a right? In my opinion, self-actualization is kind of a right in the same way society considers the pursuit of happiness a right. Everyone should have the right to try to be happy, as long as it doesn't harm others. Life goes on even if you're unhappy, but we all seek to be happy, and for the most part, everyone is entitled to try to achieve happiness. In the same sense, we all ideally strive to be the best that we can be and to reach our potential, but life goes on even if we don't. Yet, we should all be allowed to improve ourselves and grow, as long as it doesn't negatively affect other people's lives.