I think IAU is equivalent to Vopěnka's principle.  For the other direction, assume 
Vopěnka's principle fails. Then there is a proper class of structures (WLOG graphs), none of which embeds into any other.  Because this is a proper class, there is an injection of $\mathcal{L}_{\infty,\infty}$ into it.  In other words, for each sentence $\sigma \in \mathcal{L}_{\infty,\infty}$ we may choose a structure $\mathcal{M}_\sigma$ in such a way that for any two distinct sentences $\sigma,\tau \in \mathcal{L}_{\infty,\infty}$, neither $\mathcal{M}_\sigma$ nor $\mathcal{M}_\tau$ embeds into the other.

Now we define the "diagonal" class of structures $D = \{\mathcal{M}_\sigma \mid \sigma \in \mathcal{L}_{\infty,\infty} \wedge \mathcal{M}_\sigma \not\models \sigma\}$.  I claim that its upward closure $D\mathord{\uparrow}$ does not have the form $\operatorname{Mod}(\tau)$ for any sentence $\tau \in \mathcal{L}_{\infty,\infty}$.  This is because $\mathcal{M}_\tau \in D\mathord{\uparrow} \iff \mathcal{M}_\tau \in D \iff \mathcal{M}_\tau \not\models \tau \iff \mathcal{M}_\tau \notin \operatorname{Mod}(\tau)$.

The key point is that $\mathcal{M}_\tau \in D\mathord{\uparrow}$ implies $\mathcal{M}_\tau \in D$ because no other structure $\mathcal{M}_\sigma \in D$ can embed into $\mathcal{M}_\tau$.