I don't know a reference, but I think that we can use a theorem of Moser to get what you want.

Start with any triangulation $\mathcal{T}$ of $M$, whose cardinal is denoted by $k$ and denote by $\omega$ the Riemannian volume form. There is a positive smooth function $f$ such that $\int_M f\omega=\mathrm{vol}(M)$ (i.e. $f$ has average $1$) and such that for all $T\in\mathcal{T}$: $$\int_T f\omega = \frac{\mathrm{vol}(M)}{k}$$ (simply adjust $f$ in the interior of each simplex).

Now, Moser proved in 1965 (answering a question of a MO user by the way) that, since $f\omega$ and $\omega$ have the same total volume, there is a diffeomorphism $\phi:M\to M$ such that $\phi_*(f\omega)=\omega$. Then by the change of variable formula, $$\int_{\phi(T)} 1\,\omega= \int_T f\,\omega = \frac{\mathrm{vol}(M)}{k}$$ so that the collection $\{\phi(T)\}$ is a triangulation of $M$ with all simplices of the same volume.

I don't know a reference, but I think that we can use a theorem of Moser to get what you want.

Start with any triangulation $\mathcal{T}$ of $M$, whose cardinal is denoted by $k$ and denote by $\omega$ the Riemannian volume form. There is a positive smooth function $f$ such that $\int_M f\omega=\mathrm{vol}(M)$ (i.e. $f$ has average $1$) and such that for all $T\in\mathcal{T}$: $$\int_T f\omega = \frac{\mathrm{vol}(M)}{k}$$ (simply adjust $f$ in the interior of each simplex).

Now, Moser proved in 1965 (answering a question of a MO user by the way) that, since $f\omega$ and $\omega$ have the same total volume, there is a diffeomorphism $\phi:M\to M$ such that $\phi_*(f\omega)=\omega$. Then by the change of variable formula, $$\int_{\phi(T)} 1\,\omega= \int_T f\,\omega = \frac{\mathrm{vol}(M)}{k}$$ so that the collection $\{\phi(T)\}$ is a triangulation of $M$ with all simplices of the same volume.