The following theorem is a corollary of the topological invariance of domain:
Let $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ be a continuous and injective function. If $|f(x)|\rightarrow \infty$ as $|x|\rightarrow \infty$, then $f$ is surjective.
As I understand, every octonion $q \in \mathbb{O}$ can be represented as a matrix that has an inverse. Thus, every octonion polynomial $p(q)$ is a non-singular linear map from $\mathbb{R^8}$ to $\mathbb{R^8}$. If it holds that $|p(q)|\rightarrow \infty$ as $|q|\rightarrow \infty$, then can we infer from the above theorem that $0\in p(\mathbb{O})$ ?
It seems that you are confusing the fact that $p(q)$ can be interpreted as a bijective linear map $\mathbb{R}^8\to \mathbb{R}^8$, and the fact that $p:\mathbb{O}\to \mathbb{O}$ is injective.
To apply the theorem you quote, you would need $p$ to be injective, but it is not the case at all (it will usually have infinite fibers).