In classical physics every state evolved from the previous state. This is called determinism and it means that every state was caused by a sequence of previous states. By this principle you should be able to calculate from a precisely known state backwards and forward in time and therefore predict past and future. So in that sense every state has a cause and is a cause.
But quantum mechanics showed us that the universe is not deterministic but probabilistic. But even there we have equations describing change over time (Schrödinger equation), but they don't describe the change of the state of a system directly but the probability of measurement outcomes which can change over time. But even then the current quantum state is determined by previous quantum states.
So you could argue that every law of physics using time t as a variable implies a causation, but the causation only becomes apparent, when you look at different points in time.And you basically have to do consider this as causation, because if you don't, causation is non-existent anywhere.
As to what Poincaré might have meant, I would think it points to the actual idea of a force. Not only is there no such isolated thing, there were even forces that had to give way to better explanations, most impactful probably gravitation, which was replaced by the curvature of spacetime. So a force is merely a concept invoked to express a change to a system from the outside. An electron might change its direction in the presence of a magnetic field, but we don't even know what a magnetic field really is, since it has no other effect than affecting electric charges. We kind of standardized the unit Newton, but we get to it in many different ways with unrelated sets of units.
And as for causation we are unable to explain how any causation might even take place. On the quantum level it is often called the "collapse of the wave function" when something actually causes something else. Better is maybe the expression "update of the wave function". This happens when something like a measurement takes place, meaning any interference with a quantum system, including observation. The system seems to basically snap into another state and so far we have no useful model describing this.
Some additional ideas are on https://en.wikipedia.org/wiki/Causality_(physics)