Short answer: no. .NET doesn't need to walk up the scope chain to find the variables.
Long answer:
Start with this example:
static Func<string> CaptureArgs(int a, int b)
{
    return () => String.Format("a = {0}, b = {1}", a, b);
}
static void Main(string[] args)
{
    Func<string> f = CaptureArgs(5, 10);
    Console.WriteLine("f(): {0}", f());
    // prints f(): a = 5, b = 10
}
In the CaptureArgs method, a and b exist on the stack. Intuitively, if we reference the variables in an anonymous function, return the function and popping the stack frame should remove a and b from memory. (This is called the upward funargs problem).
C# doesn't suffer from the upwards funargs problem because, behind the scenes, an anonymous function is just fancy syntax sugar over a compiler-generated class. The C# code above turns into:
private sealed class <>c__DisplayClass1
{
    // Fields
    public int a;
    public int b;
    // Methods
    public string <CaptureArgs>b__0()
    {
        return string.Format("a = {0}, b = {1}", this.a, this.b);
    }
}
The compiler creates and returns a new instance of <>c__DisplayClass1, initializes its a and b fields from the a and b passed into the CaptureArgs method (this effectively copies a and b from the stack to fields existing on the heap), and returns it to the caller. Calling f() is really a call to <>c__DisplayClass1.<CaptureArgs>b__0().
Since the a and b referenced in <CaptureArgs>b__0 are vanilla fields, they can be referenced directly by the delegate, they don't require any special sort of scope chaining rules.