This is too long for a comment. (But I have now added an answer to the original question.)

As André and Ryan have hinted at in the comments, what you have is a principal fibre bundle $G_2 \to \operatorname{Spin}(7) \stackrel{\pi}{\to} S^7$.  Fibre bundles are locally trivial, so of course around every point of the 7-sphere there is a neighbourhood $U$ so that $\pi^{-1}U \cong U \times G_2$.  The diffeomorphism depends on a choice of section: a way to assign to every point in $U$ a unique element of $\operatorname{Spin}(7)$.

What I know how to describe in terms of octonions (i.e., Cayley numbers) is the above fibre bundle, and maybe this helps you.

Let $\mathbb{O}$ denote the Cayley numbers.  They form an 8-dimensional vector space with basis $e_1,\dots,e_8$, where $e_1,\dots,e_7$ are imaginary units and $e_8 = 1$.  Let $L_i$ denote left multiplication by the imaginary unit $e_i$, for $i=1,\dots, 7$.  The $L_i$ are endomorphisms of $\mathbb{O}$ which obey the Clifford relations
$$ L_i \circ L_j + L_j \circ L_i = \begin{cases} - \operatorname{id} & i=j \cr 0 & i\neq j \end{cases}$$
whence they define a linear representation of the Clifford algebra $C\ell(7)$ on $\mathbb{O}$. (You could also use right multiplication and this would give the other inequivalent Clifford module of $C\ell(7)$.  Both Clifford modules turn out to be equivalent under the spin group.)

The spin group $\operatorname{Spin}(7)$ naturally lives inside $C\ell(7)$, whence we also have a linear representation of $\operatorname{Spin}(7)$ on $\mathbb{O}$.  This is nothing but the spinor representation $\operatorname{Spin}(7) \to \operatorname{SO}(\mathbb{O})$.  The orbit of $1 \in \mathbb{O}$ under $\operatorname{Spin}(7)$ is the sphere of unit octonions, which we can identify with $S^7$.  The stabiliser of $1$ in $\operatorname{Spin}(7)$ is precisely a $G_2$ subgroup.  This then gives the principal bundle
$$G_2 \to \operatorname{Spin}(7) \stackrel{\pi}{\to} S^7$$
with $\pi(g) = g \cdot 1$.

**Added**

Here is how to construct a local section of $S^7 \to
\operatorname{Spin}(7)$ near $1 \in S^7 \subset\mathbb{O}$ using
octonions.

Under the identification of $S^7$ with the unit-norm octonions, the
tangent space $T_1 S^7$ is identified with the imaginary octonions
$\operatorname{Im}(\mathbb{O})$.  Let $\varphi$ denote the
$G_2$-invariant 3-form on $\operatorname{Im}(\mathbb{O})$.  The image
of the map $\operatorname{Im}(\mathbb{O}) \to
\Lambda^2\operatorname{Im}(\mathbb{O})$ which sends $\xi$ to
$\iota_\xi \varphi$ is a $G_2$-invariant complement of $\mathfrak{g}_2
\subset \mathfrak{spin}(7)$, and we can exponentiate them to elements
of $\operatorname{Spin}(7)$ near the identity.  Acting on $1 \in S^7$ 
we coordinatise a neighbourhood of $1$.

More explicitly, in case this is useful, the components
$\varphi_{ijk}$ of $\varphi$ relative to the basis $e_i$ above are
given by octonion multiplication as follows:
$$
e_i e_j = -\delta_{ij} 1 + \sum_{k=1}^7 \varphi_{ijk} e_k
$$
Then the point of $S^7$ with coordinates $\xi = \sum_{i=1}^7 \xi_i e_i \in
\operatorname{Im}(\mathbb{O})$, is given by
$$
\exp\left(\tfrac12 \sum_{i,j,k=1}^7\xi_i \varphi_{ijk} L_j\circ L_k\right)
\cdot 1
$$
where the $L_j$ endomorphisms were defined above.