At the top of studio pro, simply press the “Publish” button. That will deploy the project to the sandbox.
Please refer this blog
Hi Amruta Deore,
By clicking Publish you can deploy your app to sandbox. But you need to be Technical contact. If you’re not a technical contact person you can’t able to publish.
In the Mendix Studio, press the button ‘Publish’
In the Mendix Studio Pro, press the ‘Publish’-button
Seems the online services are disabled while creating the app, which you are using.
Due to the same reason the "Publish" button is disabled.