If $M$ is a $C^{1}$ manifold and $\omega_i$, $i=1,2$ are two continuous volume forms with the same mass, does there exist a $C^1$ diffeomorphism sending one to the other?
The answer is no. The equivalent problem is: given a positive and continuous function $g$, can we find a diffeomorphism with the Jacobian equal to $g$, $\det Df=g$? A counterexample is Theorem 1.2 in [1].
If $M$ is a $C^{1,\alpha}$ manifold and $\omega_i$, $i=1,2$ are two $C^\alpha$ volume forms with the same mass, does there exist a $C^{1,\alpha}$ diffeomorphism sending one the other?
The answer is yes, at least locally (see also a comment of alvarezpaiva). Whether the result has a global version on manifolds, I do not know. This is Theorem 1 in [3]. For a comprehensive treatment of related results, see [2].
Theorem. Let $k\geq 0$ and $\alpha\in (0,1)$. If $\Omega\subset\mathbb{R}^n$ is abounded domain with $C^{k+3,\alpha}$ boundary and $\omega\in C^{k,\alpha}$ is a volume form such that $\int_\Omega\omega=|\Omega|$, then there is a diffeomorphism $\varphi:\Omega\to\Omega$ that is identity on the boundary, $\varphi,\varphi^{-1}\in C^{k+1,\alpha}(\bar{\Omega})$ and such that $\varphi^*\omega=dx_1\wedge\ldots\wedge dx_n$.
[1] D. Burago, B. Kleiner, Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps. Geom. Funct. Anal. 8 (1998), 273–282. (MathSciNet review.)
[2] G. Csató, B. Dacorogna, O. Kneuss, The pullback equation for differential forms. Progress in Nonlinear Differential Equations and their Applications, 83. Birkhäuser/Springer, New York, 2012.
[3] B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 1–26. (MathSciNet review.)