There are different viewpoints in the literature about what constitutes an "affine Coxeter group" and its "Coxeter complex", so it would be helpful to specify what source you are following here. Coxeter's own approach was highly geometric, while Bourbaki and others have generalized the theory while making it more combinatorial.

I'm taking Bourbaki Groupes et algebres de Lie (Chap. V along with VI.2) as a basic reference, though my own book is organized differently (see Chapter 4 especially for affine Weyl groups). Here the defining translations come from the coroot lattice of a root system, though there's also a Langlands dual version using the root lattice which plays a bigger role in modular representation theory.

In any case, you get infinitely many reflecting hyperplanes, whose complement has (open) connected components called by Bourbaki alcoves (what you call a "chamber"). To get an alcove to be a Euclidean simplex, you need to start with an irreducible Coxeter system (or connected Coxeter graph). Here the graph is the extended Coxeter graph of the usual type $A-G$ graph for an irreducible finite root system with origin 0 and Weyl chambers comprising the connected components of the complement of the union of the finitely many root hyperplanes.

In this set-up, you'd fix simple roots and thus a dominant Weyl chamber. Then there is a unique alcove in this chamber having 0 as a vertex. Its other vertices are just the fundamental weights divided by the corresponding coefficient of the highest root when written as a $\mathbb{Z}^+$-linear combination of simple roots. For type $A_\ell$ only, all these coefficients are 1, but otherwise the vertices of the "fundamental" alcove are not all in the underlying coroot lattice. (This has to be visualized case-by-case using Bourbaki's tables.)

The answer to your question depends then on exactly what you mean by vertices of the Coxeter complex.

There are different viewpoints in the literature about what constitutes an "affine Coxeter group" and its "Coxeter complex", so it would be helpful to specify what source you are following here. Coxeter's own approach was highly geometric, while Bourbaki and others have generalized the theory while making it more combinatorial.

I'm taking Bourbaki Groupes et algebres de Lie (Chap. V along with VI.2) as a basic reference, though my own book is organized differently (see Chapter 4 especially for affine Weyl groups). Here the defining translations come from the coroot lattice of a root system, though there's also a Langlands dual version using the root lattice which plays a bigger role in modular representation theory.

In any case, you get infinitely many reflecting hyperplanes, whose complement has (open) connected components called by Bourbaki alcoves (what you call a "chamber"). To get an alcove to be a Euclidean simplex, you need to start with an irreducible Coxeter system (or connected Coxeter graph). Here the graph is the extended Dynkin diagram of the usual type $A-G$ diagram for an irreducible finite root system with origin 0 and Weyl chambers comprising the connected components of the complement of the union of the finitely many root hyperplanes.

In this set-up, you'd fix simple roots and thus a dominant Weyl chamber. Then there is a unique alcove in this chamber having 0 as a vertex. Its other vertices are just the fundamental (co)weights divided by the corresponding coefficient of the highest root when written as a $\mathbb{Z}^+$-linear combination of simple roots. For type $A_\ell$ only, all these coefficients are 1, but otherwise the vertices of the "fundamental" alcove are not all in the underlying coroot lattice. (This has to be visualized case-by-case using Bourbaki's tables.)

The answer to your question depends then on exactly what you mean by vertices of the Coxeter complex.