Currently neither Grammarly nor the LanguageTool browser add-on are directly supported on the new Overleaf platform, however if you are using the older Overleaf v1, you will be able to use the LanguageTool add-on as detailed below.
The LanguageTool browser add-on/plugin works with the Overleaf v1 editor. This provides a language and grammar checker which allows you to check your text for grammatical errors. You can download/install the add-on for Firefox or Chrome.
After installing the add-on, highlight some text in the Overleaf v1 editor which you'd like to check, and then click on the LT button in your browser's tool bar. The add-on will then list possible problems it found in the selected text.
Please note that, as mentioned at the top of this article, the LanguageTool add-on is currently not supported on the new Overleaf platform.