You could just require IContainer dependency in your action filter's constructor. If it's not registered automatically by StructureMap, you could register it with:
For<IContainer>().Use<Container>();
Edit
Option 1: Couldn't you just use something like:
GlobalConfiguration.Configuration.ServiceResolver.GetService(...)
// or (not sure what would be the right syntax)
GlobalConfiguration.Configuration.ServiceResolver.Current.GetService(...)
There got to be some way of retrieving the current service resolver once you set it.
Option 2: With regular MVC you can get the current resolver like this:
DependencyResolver.Current
And use it like this:
DependencyResolver.Current.GetService()
Looks like WebAPI doesn't use DependencyResolver, but according to this blog post, you could set it like this:
DependencyResolver.SetResolver(new StructureMapDependencyResolver(container));
// this override is needed because WebAPI is not using DependencyResolver to build controllers
GlobalConfiguration.Configuration.ServiceResolver.SetResolver(
DependencyResolver.Current.GetService,
DependencyResolver.Current.GetServices);
Now try to use DependencyResolver.Current from your action filter.
Option 3: Use ObjectFactory.GetInstance directly - Probably not the best idea in MVC project, since it should already be encapsulated in IDependencyResolver instance.