I built a Service Fabric Application with two services:
Yes - you can use reliable collections in an asp.net service. This sample is equal to what you are building: https://github.com/Azure-Samples/service-fabric-dotnet-quickstart it contains an asp.net core stateful service.
Main things:
Now your controllers will have access to the singleton StateManager:
public MyDataController(IReliableStateManager stateManager)
{
this.stateManager = stateManager;
}