In the Export tab, when clicking the Export button, a window pops up where you can specify the export destination.
Is it possible to write a plugin to customize this window? I want to let the user specify a cloud service instead of storing locally.
No, unfortunately this windows is not open to making integrations. You would be able to create a different export functionality as a plugin but it would then require that you create your own dialogs and similar.
The Smart Client Export sample could be a start or inspiration if you want to develop export functionality to replace what is currently in the Smart Client.