I was informed from Zeplin's crew that this is fixed in Zeplin's v7.5 update. Confirming that it works now. XD's export has no errors and it exports grid too.
This is a known issue happening with the latest version of Adobe XD and Zeplin's legacy integration—affecting Windows users at the moment. Our hands are sadly a bit tied here as the issue happens on Adobe XD's side. We're in contact with the XD team to solve this issue.
The grid layouts are currently not supported by Adobe XD's plugin API. That's why it's not possible to publish the designs with their grid layouts through the plugin. We're also in contact with Adobe XD to resolve this. If you'd like to continue publishing your designs with their grid layout, I'd recommend you downgrade Adobe XD to its previous version and try publishing your designs through the legacy integration.
You can install Zeplin's plugin and go to menu PLUGINS > ZEPLIN FOR XD > EXPORT SELECTED... However, it has not identical possibilities like native XD export (to Zeplin). For example, to me very important, with plugin you cannot export grid (if you using it), while XD's export can do it. I think that Adobe has to fix its own export module.
I'm also having the same problem since I upgraded XD to v49.012.14.
When I try to export (using File > Export), it says that Adobe XD CC is not running (but it is), and I have to launch it again.
Regarding workaround. It is important to say that my Zeplin plugin works, but I prefer XD's export, as it exports grid which is very important for my developers (while the plugin doesn't).