Entanglement is a cornerstone of modern quantum many-body theory. Experimentally, however, it is extremely difficult to observe, especially multipartite entanglement. Here, we show that the quantum Fisher information, a witness for genuinely multipartite entanglement, can be measured generically in thermal ensembles through the dynamical susceptibility. This connection, which is independent of microscopic details, makes multipartite entanglement accessible to experiments using standard techniques. Moreover, it allows us to identify a class of quantum phase transitions with strong universal scaling, leading to divergent multipartite entanglement. We illustrate our framework with several numerical examples, all realizable in ultracold-atom experiments.