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 clear -- just permute $(g_1, g_2, g_3)$ via the symmetric group $S_3$.
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.