In practice, there is no concept as ASP.NET Web API support in terms of hosting. What you should really be doing is to install the ASP.NET Web API NuGet package into your project and the assemblies you get through this package are bin deployable.
In terms of hosting, What you need to be looking for on your hosting options is the support for IIS 7 or higher (I don't think IIS 6 is supported but I am not sure), .NET 4.0 or higher (the higher is .NET 4.5 now) and Integrated Application Pool availability. If you have those, an ASP.NET Web API application should run without any problems under IIS.
If you ask the hosting provider for support for ASP.NET Web API, it is likely that they won't know the answer. Instead, you can ask for the features that I listed above.
