Can I add LaTeX packages to MathJax?

Is there a way of using LaTeX packages with MathJax? If so, how would I do this?

1. First the StackExchange site must support MathJax.

2. You need to use the correct syntax for the particular site .

   Some sites require a backslash before the dollar sign, because when people try to write "it costs $10 shipping each time" they end up with "it costs $10 shippingeachtime", so to avoid excessive questions about why that happens the dollar sign must be escaped to use MathJax on some sites.

3. You need to use $\require{\packagename}$ somewhere on the page, and it affects the whole page (including other people's posts).

4. The site in question needs to support the extension. The official (but incomplete) list is here: "MathJax third-party extensions".

The most complete MathJax Tutorial and Links answer is over at QC.meta.SE.

Specific to your question, "ytableau" didn't work on the site where I tested these examples (physics.SE) but the other packages I demonstrate do work:

$\require{\AMScd}$

$$\begin{CD}
A @<<< B @>>> C\\
@. @| @AAA\\
@. D @= E
\end{CD}$$

$\require{\mhchem}$

$$\ce{Zn^2+  <=>[+ 2OH-][+ 2H+]  
$\underset{\text{zinc hydroxide}}{\ce{Zn(OH)2 v}}
$  <=>[+ 2OH-][+ 2H+] 
$\underset{\text{tetrahydroxozincate(II)}}{\ce{[Zn(OH)4]^2-}}$}$$

_{The mhchem example was obtained from the MathJax-mhchem webpage.}

Test a short example to determine if and how it works. Worst come to worst you can compose on a supporting site, capture and crop a screenshot, then enclose the MathJax in HTML comments <!-- comment --> and upload the screenshot - if in the future MathJax is enabled the code will be available in your question or answer within the HTML comment.

Is there a way of using LaTeX packages with MathJax?

No. MathJax isn't LaTeX; it processes input that is LaTeX-like (as close as it reasonably can be to true LaTeX), but the way it works under the hood is very different. One difference in particular is that MathJax only implements the math-mode macros, not all the other text-mode or general processing macros and control sequences. So most of the tools used by LaTeX packages aren't there. That means you can not simply drop a LaTeX package into MathJax. The packages that MathJax supports have to be re-implemented in javascript, which is difficult for non-trivial packages (like the one you cite).