For a conservative potential, there is no causality in the least action principle. Feynman has explained this a bit in his The Character of Physical Law Lecture 2 - The Relation of Mathematics to Physics.
[Regarding three equivalent ways to describe the gravitational law: field, action at a distance, and the least action principle] ... Now if you calculate this quantity [Lagrangian] for this route and for another route, you'll get, of course, different numbers for the answer. But there's one route which gives the least possible number for that, and that's the route that the particle takes. Now we're describing the actual motion, the ellipse, by saying something about the whole curve. We have lost the idea of causality, that the particle is here. It sees the pull. It moves to [t]here.
Instead of that, in some grand fashion, it smells all the curves around here, all the possibilities,
and decides which one to take. So this is an example of the wide range of beautiful ways of describing nature and that when people talk that nature must have causality, well, you could talk about it this way [LEAST ACTION]. Nature must be stated in terms of a minimum principle. Well, you can talk about it this way [FIELD]. Nature must have a local field. Well, you can do that [ACTION AT A DISTANCE] and so on.
And the question is, which one is right? Now if these various alternatives are mathematically not exactly equivalent and if for certain ones there will be different consequences than for others, then it's a very-- that's perfectly all right then because we have only to do the experiments to find out which way nature actually chooses to do it. Mostly, people come along, and they argue philosophically. They like this one better than that one, but we have learned from much experience that all intuitions about what nature's going to do philosophically fails. It never works. [......]
However, keep in mind that the uniqueness theorem is not always guaranteed. Crazy things can happen when the uniqueness theorem is not held.
For some non-conservative potential, it is possible for the least action principle to give infinite possible paths. Classically, the particle will take one path - out of the many paths - that can only be determined by both initial position and initial momentum. (All of the paths obey the Newtonian equation of motion.) In this case, there is causality in the least action principle (since you really need initial conditions to specify which path is taken, and different initial conditions give different one path). $^\dagger$
$\dagger$ : Given a "crazy enough" potential field, even Newtonian mechanics turns out to be non-deterministic or not causal.