For a long time this would not have been possible, but there is now a Beta API for Grammarly. I still don't know if it will be possible but we have applied for access to the SDK and if they respond appropriately we will take a look and see whether we can deliver this or not.
No. I didn't even receive an email from them, so I have applied again. Even if they do respond positively we probably won't look at this for some months.
I am a premium subscription user of Grammarly, which I use as part of my QA prior to delivery of a translation into English. Inspired by the above, I have just written to Grammarly with a request to make integration into Trados Studio available (with a requirement for subscription to make full use of all facilities).
Maybe if other users of Grammarly make similar requests, they might see some advantage in adding this facility.