This is something I've looked into before, and I had a very difficult time getting it to function. Getting a spell checker installed on the site requires installing a separate English language dictionary (a very specific one at that). Finding the specific dictionary and figuring out how to install it was a bit out of my wheelhouse as all the instructions were in incomprehensible tech jargon that I am currently not educated enough to decipher.
It can be something I look into for the future, but it will require a bit of studying on my part.
As for whether the site/layout is compatible with specific browser add ons, that is not something I can accurately tell you. Without knowing which specific add on you're using, I have no way of knowing whether something you're using is incompatible with the Xenforo software or a glitch on the side of your browser.
None of the layouts have their own spellcheck as that would be something installed onto the server, making it available for the entire site. And as previously mentioned, I cannot test it myself, unfortunately.
In the meantime, I would recommend doing some testing on your own (trying different site layouts, using a different browser add on, etc). This will help you narrow down whether this is a browser issue or a Xenforo issue, which is information we may be able to utilize in the future.