Yes, normally you can generate a GRBL-like G-code with Solidworks Cam.
You have to find a GRBL post processor compatible with solidworks, with a quick query on a search engine like "Solidworks GRBL post processor" you should find some. Check and simulate on your CNC control software before sending it out for machining, but normally it will be good.
That's precisely what I did 2 days ago, I was planning to update this topic.
By using the CTL file of the zip of this http://www.restrictedayerspace.net/cnc/grbl-postprocessor-for-solidworks/ link, I was able to solve a problem of edge management, with very particular behavior, and also to save myself from now on a few code edits to delete lines not recognized by the GRBL controller.
This test proved to be effective (to be confirmed with future projects) in answering the question of generating a compatible G-Code.
However, it does not seem possible to directly control a CNC from Solidworks. Unless there is an add-in for that?