No economist regardless of training or methodological background would deny that people "act". It might be the only axiom all social scientists have to accept as a prerequisite to the study of human behavior. That people "act" is not really a point of contention among economists. The real disagreement between Austrians and other schools of thought isn't that people act, it's how and why they act the way they do.
The axiom of action is one the foundational assumptions on which most Austrian theories are based on, and it's probably a perfectly valid starting place. But you can only take such an initial assumption so far before you're forced to start making heroic leaps in your successive assumptions and end up reasoning your way into absurdity. In Mises' case when you go from a fairly uncontroversial truism like "action is an attempt to substitute a more satisfactory state of affairs for a less satisfactory one" to dubious claims like "deflation and credit restriction never played a noticeable role in economic history" you've obviously taken a logical misstep somewhere along the road. If your conclusion doesn't follow from the premise your deduction is invalid and it's time to go back to the drawing board.
Therin lies one of my problems with the Austrian methodology. The Austrians are so convinced of the validity of their own axioms and what should logically follow, that they assume to even bother testing their predictions against the reality of the observable world would just be a waste of time. Sure you can try, but doing so will only confirm their self-evident truth and make you look silly, so why bother? Fair enough. But just for kicks let's say we want to put one of Austrians' "perfect" deductions to the test anyway, Austrian Business Cycle Theory for example. We gather our data, analyze it, and compare it to what theory would predict only to find that it totally fails to accurately model the reality of human behavior. People are acting, but they're not acting how the ABCT would expect them to. What now?
Well when reality contradicts theory, most economists would agree that we made a mistake somewhere along the road and it's the theory that needs to be thrown out, not reality. But the Austrians do just the opposite. When the real world contradicts their simplistic assumptions about what must follow, given the axiom of action, they simply distort reality to fit what their over-exerted reasoning should've predicted. This axiomatic fundamentalism and an inability to admit to flaws when their theories fail to stand up to facts and evidence is one of the reasons scholars of the Austrian tradition haven't made any worthwhile contributions to modern economics in decades and the reason mainstream economists rightly ignore them.