This isn't a complete answer. In general, suppose $X \to BG$ is the classifying map of a principal $G$-bundle and we want to describe the obstructions to reducing it to a principal $H$-bundle, or equivalently to lifting it (up to homotopy) to a map $X \to BH$. Under mild hypotheses, which in particular hold in this case, the homotopy fiber of the map $BH \to BG$ is $G/H$, and the question of reduction of the structure group is equivalent to the question of finding a section (up to homotopy) of the $G/H$-bundle associated to the classifying map $X \to BG$.
This question can be attacked using a relative version of the theory of Whitehead towers taking place in local systems of spaces over $X$. Assume for simplicity that $X$ is connected. In general, given a local system of spaces over $X$ with fiber $F$, locally a choice of section up to homotopy amounts to a choice of basepoint in $F$. Globally the first obstruction is that the connected components $\pi_0(F)$ organize into a local system of sets over $X$, and we need to trivialize this local system. This step can be ignored if $F$ is connected.
Given such a trivialization, the next obstruction is that the fundamental groups $\pi_1(F)$ (based at the basepoints given by the choice of trivialization) organize into a local system of homotopy $1$-types $B \pi_1(F)$ over $X$, and we need to trivialize this local system. This step can be ignored if $F$ is simply connected.
From here, the higher homotopy groups $\pi_k(F)$ organize into local systems of homotopy $k$-types $B^k \pi_k(F)$ (dependent on a choice of trivialization of the previous local system), and we need to trivialize them. These local systems are classified by classes in cohomology with local coefficients $H^{k+1}(X, \pi_k(F))$ (only well-defined given a choice of trivialization of the previous classes) and so are trivializable iff the corresponding classes vanish. If $X$ is an $n$-manifold then we only need to worry about the classes where $k \le n - 1$.
Now let's specialize to the case that $G = Spin(8), H = SU(4)$. The spaces $BSU(4)$ and $BSpin(8)$ are both $3$-connected, and the natural map $BSU(4) \to BSpin(8)$ induces an isomorphism on $\pi_4$ (the corresponding map on $H^4$ sends the first fractional Pontryagin class $\frac{p_1}{2}$ to $- c_2$) and on $\pi_5$ (which vanishes for both spaces). We also know that $\pi_6 BSpin(8) = \pi_7 BSpin(8) = 0$ (this is in the stable range so is handled by Bott periodicity). An inspection of the long exact sequence in homotopy then gives that the homotopy fiber
$$F = Spin(8) / SU(4)$$
is $5$-connected and that $\pi_6 F \cong \mathbb{Z}$. We only need to worry about two characteristic classes, one in $H^7(X, \pi_6 F) \cong H^7(X, \mathbb{Z})$ and one in $H^8(X, \pi_7 F)$.
At this point I have to admit that I don't know what these characteristic classes are. I have a guess: the one in $H^7(X, \mathbb{Z})$ may be the integral Stiefel-Whitney class $\beta w_6 = W_7$; it vanishes iff $w_6$ is the reduction of an integral class $c_3$. But I really don't know what this class in $H^8(X, \pi_7 F)$ is.
