Following this guide from Microsoft, it seems fairly straightforward to migrate web and worker roles to service fabric. This picture specifically shows how you can still use Azure Storage from Service Fabric apps.
Since web roles must be .net core on service fabric, this requires that the storage sdk support .net core but I have been unable to find libraries that are compatible. There seems to be a storage preview package that does support netstandard1.6 but it's dependencies do not.
Has anyone gotten this to work? Basically, I'm looking for a storage package or configuration that will work with .net core so that I can use Azure storage from a asp mvc core app.
ASP.NET Core web applications don't actually need to be compiled against .NET Core - these are two separate things. In fact the Service Fabric SDK itself is still full .NET Framework. The terminology is confusing (we're bad at naming things).
"ASP.NET Core" is the name of the latest ASP.NET product that Service Fabric supports.
You can build an ASP.NET Core application against either .NET Core or .NET Framework. In Visual Studio, if you create a new project under C# -> Web, you'll see ASP.NET Core options for both .NET Core and .NET Framework.
On Service Fabric, currently, you should use ASP.NET Core built against .NET Framework, not .NET Core. In Visual Studio again, if you create a new project under C# -> Cloud -> Service Fabric Application, you'll have the option of adding an ASP.NET Core service - this will be created using full .NET Framework.