According to the introduction in

Cooper, D.; Thurston, W. P., Triangulating 3-manifolds using 5 vertex link types, Topology 27, No. 1, 23-25 (1988). ZBL0656.57004.

It is known that, for any dimension $n$, there is a finite set of link types such that every $n$-manifold has a triangulation in which the link of each vertex is in this set.

(I assume the statement is about PL manifolds.)

What is a proof of or a reference to this known result?

According to the introduction in

Cooper, D.; Thurston, W. P., Triangulating 3-manifolds using 5 vertex link types, Topology 27, No. 1, 23-25 (1988). ZBL0656.57004.

It is known that, for any dimension $n$, there is a finite set of link types such that every $n$-manifold has a triangulation in which the link of each vertex is in this set.

(I assume the statement is about PL manifolds.)

What is a proof of or a reference to this known result?

Edit. There is a related discussion by Florian Frick here, but he only gets a bound on the valence of "ridges" (codimension 2 faces). I do not see how to generalize his sketch to faces of higher codimension. (Actually, I cannot follow his sketch.)