I have a lot of hesitations about how these models are employed in practice (abusing model selection methods to try to “learn” causal structure, implicitly defining prior models to avoid chaotic behavior in dynamical systems, etc) but the models themselves are technically within the scope of Stan. In other words they can readily be implemented as well-defined Stan programs.
Stan, however, will also not be quiet about computational problems inherent to these models that hand-written samplers might ignore (the included papers list many references to support their hand-written samplers, but they also make some common misunderstandings about how Markov chain Monte Carlo works). Although frustrating these diagnostics will allow you to investigate and understand these less-than-ideal model consequences in a way that no other method will, which makes it a rare feature.
The most subtle challenge here will be the chaotic nature of the dynamical systems. Stan’s ODE integrators will do their best, but once the dynamical systems become unstable the numerical evaluation of the states and the gradients will tend to decouple which will degrade the performance of Stan’s dynamic Hamiltonian Monte Carlo sampler.
tl;dr Yes these models can be implemented in Stan, but they may be challenging to fit. That said they’re no easier to fit with other methods and Stan is uniquely capable of guiding investigations of the fitting problems.