I saw a blogpost from Overleaf, saying that LT works with it (LanguageTool—a free browser add-on to check your grammar and spelling - Overleaf, Editor de LaTeX online).
However, when I downloaded the extension for both Firefox and Chrome, it shows as off when I am on Overleaf.
Is there any solution to that?
There’s no solution yet, but we’re aware of the issue. We’re currently considering the options of how Overleaf can be supported.
Any news on this? 
Overleaf.com is now supported starting with version 4.0 or our browser add-on.
Another update: LanguageTool currently works with the “Source (legacy)” and “Rich Text” modes of Overleaf:
We’ll try to update our add-on to work with the default source mode again.