Since I can define an Action as
Action a = async () => { };
Can I somehow determine (at run time) whether the action a is async or not?
No - at least not sensibly. async is just a source code annotation to tell the C# compiler that you really want an asynchronous function/anonymous function.
You could fetch the MethodInfo for the delegate and check whether it has an appropriate attribute applied to it. I personally wouldn't though - the need to know is a design smell. In particular, consider what would happen if you refactored most of the code out of the lambda expression into another method, then used:
Action a = () => CallMethodAsync();
At that point you don't have an async lambda, but the semantics would be the same. Why would you want any code using the delegate to behave differently?
EDIT: This code appears to work, but I would strongly recommend against it:
using System;
using System.Runtime.CompilerServices;
class Test
{
    static void Main()        
    {
        Console.WriteLine(IsThisAsync(() => {}));       // False
        Console.WriteLine(IsThisAsync(async () => {})); // True
    }
    static bool IsThisAsync(Action action)
    {
        return action.Method.IsDefined(typeof(AsyncStateMachineAttribute),
                                       false);
    }
}
                        Of course, You can do that.
private static bool IsAsyncAppliedToDelegate(Delegate d)
{
    return d.Method.GetCustomAttribute(typeof(AsyncStateMachineAttribute)) != null;
}
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