The code
\require{AMScd}\begin{CD}
0 @>>> \mathbf D(C) \\
0 @>>> M
\end{CD}
produces $$\require{AMScd}\begin{CD} 0 @>>> \mathbf D(C) \\ 0 @>>> M \end{CD}$$
To make sure we're seeing the same thing, when I preview the page, it shows up thus for me, with the arrow in the second row aligned with the $\mathbf D(C)$ in the first row:
This happens with Firefox, Safari, and Chrome, with all renderers (at least on Safari—I didn't try them all on each browser), in macOS 13.1.
When compiled using TeX (as opposed to MathJax), the result of the code, mutatis mutandis, has the expected alignment of the two arrows:
Including vertical arrows restores the expected behaviour:
\begin{CD}
0 @>>> \mathbf D(C) \\
@| @VVV \\
0 @>>> M
\end{CD}
$$\begin{CD} 0 @>>> \mathbf D(C) \\ @| @VVV \\ 0 @>>> M \end{CD}$$
even if the arrows are ‘empty’:
\begin{CD}
0 @>>> \mathbf D(C) \\
@. @. \\
0 @>>> M
\end{CD}
$$\begin{CD} 0 @>>> \mathbf D(C) \\ @. @. \\ 0 @>>> M \end{CD}$$
As @EmilJeřábek points out, changing \\ to \\\\ also fixes the problem:
\begin{CD}
0 @>>> \mathbf D(C) \\\\
0 @>>> M
\end{CD}
$$\begin{CD} 0 @>>> \mathbf D(C) \\\\ 0 @>>> M \end{CD}$$
I think that this is because \\\\ is interpreted essentially as \\{}\\ ; that is, all that's needed is a non-horizontal row between two ‘horizontal’ rows, even if it's empty.
\begin{align}...\end{align}). It’s possible that the exception only applies to specific MathJax environments, and CD is missing on the list. $\endgroup$\\; your comment inspired me to test, and adding in an interstitial row of vertical arrows, even if ‘empty’, restores the expected behaviour. (EDIT: In fact, that seems to be what your solution is doing, as indicated by the extra vertical space it introduces:\\\\seems to be effectively\\{}\\.) I have edited accordingly. $\endgroup$