Is the \begin{align} function available for the MaTeX package?

Use aligned from amsmath (I do not think align or align* works in Matex)

Needs["MaTeX`"]
SetOptions[MaTeX,"Preamble"->{"\\usepackage{amssymb,amsmath,latexsym,amsfonts,amsthm}"}]

Then

MaTeX["
\\begin{aligned}
V&= \\int_0^H dV\\\\
&= \\int_0^H A'\\,dh\\\\
&= \\int_0^H\\frac{Ah^2}{H^2}\\,dh\\\\
&= \\frac{A}{H^2}\\left|\\frac{h^3}{3}\\right|_0^H\\\\
&= \\frac{A}{H^2}\\cdot\\frac{H^3}{3}\\\\
&= \\frac{1}{3}AH
\\end{aligned}
"]

Gives

Second example

MaTeX["
\\begin{aligned}
\\int_0^r 2\\pi r\\,dr &= 2\\pi \\int_0^r r\\,dr \\\\
&= 2\\pi\\bigg\\vert_0^r \\frac{r^2}{2} \\\\
&= \\bigg(2\\pi \\frac{r^2}{2}\\bigg)-\\bigg(2\\pi \\frac{0^2}{2}\\bigg) \\\\
&= \\frac{2\\pi r^2}{2} \\\\
&= \\boldsymbol{\\pi r^2}
\\end{aligned}
"]

Gives

MaTeX interprets TeX commands in inline math mode. Only those commands work that are allowed in inline math mode.

As @Nasser said, in this case use aligned instead of align (and make sure that the amsmath package is loaded—it is in the default "BasePreamble" option value).

This is discussed both on the webpage you linked to, and the builtin documentation. Please check the documentation (in particular the Possible Issues section) when encountering problems like this. It has answers to the most common questions.