The higher version of this statement is the following: taking the fundamental $n$-groupoid $\Pi_{\le n}(-)$, or equivalently $n$-truncating, is (higher) left adjoint to the inclusion of $n$-truncated spaces into spaces. Hence it sends homotopy colimits to homotopy colimits. This implies, for example, that if you want to compute $\pi_n$ of a homotopy colimit it suffices to remember the $n$-truncations of all of the spaces involved.
This is less helpful than it soundsisn't too useful as a computational tool because for $n \ge 2$ it's hard to compute homotopy colimits of $n$-groupoids, or equivalently $n$-truncated spaces. It's also worth pointing out that higher homotopy groups $\pi_n$ really behave nicely with respect to homotopy limits, not homotopy colimits, so for $n \ge 2$ the gap between knowing $\Pi_{\le n}(-)$ of a homotopy colimit and knowing $\pi_n(-)$ is much larger than for $n = 1$.
I also hesitate to call this a version of the Seifert-van Kampen theorem because the computational power of the Seifert-van Kampen theorem comes from the fact that it guarantees that certain homotopy pushouts can be computed as ordinary pushouts. This is a model-dependent kind of statement, whereas what I've said above is model-independent.