As Thomas says in comments, this should be possible once LLILC is out. It targets .NET Core which is what ASP.NET 5 runs on. I am not sure if the resulting runtime can be called .NET Native per say but LLILC do has plans to natively compile IL (e.g. output by Roslyn) ahead-of-time.
Another option is ASP.NET running on CoreRT by using RyuJIT as AOT compiler. This looks closer to reality today than LLILC. Have seen some experiments in compiling ASP.NET project on CoreRT but nothing that actually works.
[.NET Native makes use of UTC compiler which compiles to run on some C++ runtime (MRT - either minimal runtime or managed runtime, cant remember, also known as Native runtime). Currently the .NET Native UWP apps are windows specific. Though .NET Native and UWP are advertised under .NET Core, this could be misleading as only in debug mode UWP targets CoreCLR, in release mode it targets native runtime which is Windows specific. LLILC/CoreRT should change that.]