Is the $H$-space structure on $S^7$ associative up to homotopy?

It is not. See Theorem 1.4 of this paper by I.M. James (Trans. AMS 84 (1957), 545-558).

In particular, there exists no homotopy associative multiplication on $S^n$ unless $n=1$ or $n=3$.

There is a Proof due to Stasheff in "H-space from homotopy point of view" (Theorem 6.7). The argument is fairly simple to describe. $\def\OP{{\mathbb O\mathbf P}}$

If $S^7$ admits a homotopy associative multiplication, then one should be able to construct $\OP^3$. It would follow that $\tilde{H}^*(\OP^3;\mathbb{Z}/3)$ is generated by $u_8$ in degree $8$ with the relation that $u_8^4 =0$. It follows that, $$ P^4(u_8) = u_8^3. $$ However, $P^4 = P^1P^3$, which means the $u_8^3 = P^1x$, where $x \in \tilde{H}^{20}(\OP^3;\mathbb{Z}/3) = 0$. Thus $u_8^3 = 0$ which contradicts the existence of $\OP^3$ and consequently the existence of homotopy associative multiplication on $S^7$.

More interestingly this proof suggests that the obstruction to homotopy associative multiplication is $3$-local. This leads to the following question.

Question: Is $S^7_{(2)}$ homotopy associative? Does the reference to James work in the answer due to Jon Beardsley, addresses this question? (Tyrone's comment made me think of this question.)

(Sorry for asking a question inside an "answer".)

This response is to the question raised above whether S^7_(2), the 2-localization of the 7-sphere, can admit a homotopy associative multiplication. The answer is no. This fact was established by I.M. James. A later proof by Daciberg Goncalves, "Mod 2 homotopy associative H-spaces" in Geometric Applications of Homotopy Theory I 198-216, Edited by M.G.Barratt and M.E. Mahowald, Lecture Notes in Mathematics 657, Springer-Verlag (1978), revealed the universal nature of the obstruction. The proof involves a secondary cohomology operation and has wider implications. In the same volume, John Hubbock gave a K-theory proof. The calculation of the secondary operation is also presented in the book "Secondary Cohomology Operations" Graduate Studies in Mathematics 49 AMS, by the undersigned. The details are on pages 192-3, and 217-8.