<blockquote>
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?
</blockquote>

**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].
 

<blockquote>
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?
</blockquote>

**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,** <A HREF="https://link.springer.com/article/10.1007%2Fs000390050056"><FONT FACE="Arial">Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps</FONT></A><FONT FACE="Arial">.
*Geom. Funct. Anal.* 8 (1998), 273–282. 
(<A HREF="https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=RT&pg7=JOUR&pg8=ET&review_format=html&s4=burago%20and%20kleiner&s5=&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3&mx-pid=1616135"><FONT FACE="Arial">MathSciNet review</FONT></A><FONT FACE="Arial">.)

[2] **G. Csató, B. Dacorogna, O. Kneuss,** <A HREF="https://link.springer.com/book/10.1007/978-0-8176-8313-9"><FONT FACE="Arial">The pullback equation for differential forms</FONT></A><FONT FACE="Arial">. Progress in Nonlinear Differential Equations and their Applications, 83. Birkhäuser/Springer, New York, 2012. 
(<A HREF="https://mathscinet.ams.org/mathscinet-getitem?mr=2883631
"><FONT FACE="Arial">MathSciNet review</FONT></A><FONT FACE="Arial">.)



 


[3] **B. Dacorogna, J. Moser,** 
<A HREF="http://caa.epfl.ch/publications/22-Dacorogna-Moser1990.pdf"><FONT FACE="Arial">On a partial differential equation involving the Jacobian determinant</FONT></A><FONT FACE="Arial">. 
*Ann. Inst. H. Poincaré Anal. Non Linéaire* 7 (1990),  1–26. 
(<A HREF="https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=RT&pg7=JOUR&pg8=ET&r=1&review_format=html&s4=dacorogna%20and%20moser&s5=&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq"><FONT FACE="Arial">MathSciNet review</FONT></A><FONT FACE="Arial">.)