I have searched all Microsoft documentation and I can't find anything that explicitly states whether Visual Basic is supported by .NET Core on Linux. So, does .NET Core on Linux support Visual Basic and is there a yeoman generator for Visual Basic applications?
No, there is no Visual Basic support yet. At the very bottom of the roadmap in the "future work" section, it said at one time, that Visual Basic support was set for quarter 3 of 2016, but it is still not released as of March 2017. It is still planned.
Since ASP.NET Core is the most common thing that targets the .NET Core framework, the answer I've provided refers to that.