Here's my favorite way to answer your question. Hopefully the answer to Robert Bryant's question is "yes".

Let $A$ be the ring of octonions (the "nonsplit" octonions over ${\mathbb R}$); it comes with an involution $\alpha \mapsto \bar \alpha$, from which there is a trace $Tr(\alpha) = \alpha + \bar \alpha$ and a norm $N(\alpha) = \alpha \cdot \bar \alpha$. From this, we get a trilinear form $T: A \otimes A \otimes A \rightarrow {\mathbb R}$ given by
$$T(\alpha, \beta, \gamma) = Tr( (\alpha \beta) \gamma) = Tr(\alpha (\beta \gamma)).$$
(Multiplication is nonassociative, but the traces work out to the same result.)

The group $Spin(8)$ can be constructed as the group of triples $(g_1, g_2, g_3) \in SO(A,N)^3$ of "rotation matrices" with respect to the norm quadratic form, such that for all $(\alpha, \beta, \gamma) \in A^3$,
$$T(g_1 \alpha, g_2 \beta, g_3 \gamma) = T(\alpha, \beta, \gamma).$$

The full group of outer automorphisms is now **almost** clear -- cyclic permutations of $(g_1, g_2, g_3)$ give automorphisms of $Spin(8)$ defined above since $T(\alpha, \beta, \gamma) = T(\beta, \gamma, \alpha)$.

**Edit below to reflect comments and correspondence with Daemi, and the comment of Bryant**

The full $S_3$ action on $Spin(8)$ is obtained from cyclic permutations and the following slightly subtle action of transpositions. Let $C$ denote the main involution of $A$ (the one for which $Tr(\alpha) = \alpha + C(\alpha)$). For any $g \in SO(A,N)$, define $\bar g = C \circ g \circ C$; then $\bar g \in SO(A,N)$ as well.

The action of a transposition on $Spin(8)$ follows: for the transposition $(12)(3)$, the associated outer automorphism of $Spin(8)$ sends $(g_1, g_2, g_3)$ to $(\bar g_2, \bar g_1, g_3)$. The other transpositions act in the analogous ways.

The Jordan algebra is the exceptional Jordan algebra of 3x3 Hermitian matrices with octonion entries:
$$J = \left\lbrace
\left( \begin{array}{ccc}
a & \alpha & \bar \beta \cr
\bar \alpha & b & \gamma \cr
\beta & \bar \gamma & c \\
\end{array} \right) : \alpha, \beta, \gamma \in A, a,b,c \in {\mathbb R} \right\rbrace.$$

The group $Spin(8)$ acts on the triple of octonions $(\alpha, \beta, \gamma)$ via the natural representation from above. It acts trivially on the real numbers $a,b,c$, and this gives an action of $Spin(8)$ on the exceptional Jordan algebra. The outer automorphism group $S_3$ acts by permuting $(a,b,c)$ and $(\alpha, \beta, \gamma)$ simultaneously. Together, these give an action of $S_3 \ltimes Spin(8)$ on the exceptional Jordan algebra.

**Reference update:**

The material above can be found in my paper on $D_4$ modular forms, Amer. J. of Math. 128 (2006), 849-898.

The construction of $Spin(8)$ (over ${\mathbb Z}$) follows from Proposition 4.8 of M.-A. Knus, R. Parimala, and R. Sridharan, "On Compositions and Triality," J. reine angew. Math., 457:45–70, 1994.