You can use the detex tool to strip LaTeX commands. If you do detex file.tex, it will output to stdout. Then, you can use the diction tool to analyse your text, and suggest improvements. Putting it all together:
detex file.tex | diction -bs
This will strip the LaTeX, and pipe it into diction with suggestions and "beginner mistakes" enabled.
For Debian/Ubuntu:
sudo apt-get install diction texlive-extra-utils
This will install detex and diction, as well as several other useful tools.
Since you mentioned that you're using Emacs, there is also diction.el which provides diction integration with Emacs. You can do M-x diction-buffer, and it will take care of detexing/dehtmling and show the diction results in a separate buffer. You can even hit enter on the individual results to be taken to approximately where the phrase is.
(If you do use diction.el, my version of detex didn't have the -C flag. I'm not sure of what it was meant to do, but if you search for 'detex -C' in the file, you can add other command line options if you want. Also, I recommend replacing 'diction -L' with 'diction -bsL')
If you setup your document to use XeTeX, it works just as it should: You write \text then in the little box you enter your text in Hebrew as is. The only caveat is that LyX will display the text in the formula reversed, but in the output it is fine.
Best Answer
The closest thing is that LyX has support for ChkTeX and this can be leveraged to integrate grammar checking with Language Tool. There is, however, no continuous grammar checking (e.g., blue wavy lines under grammar mistakes). Please see here for instructions on how to set up ChkTeX in LyX to use Language Tool.
Grammarly does not have an API. You can export to plain text, or copy/paste, to use Grammarly, but that would not give you good integration with LyX.