<p>I want to connect to a USB Webcam in .NET, specifically using C#. Being new to .NET I don't know what kind of support there is in the standard libraries for doing so. I found one example on the web that copies bitmaps through the clipboard, but that seems very hacky (and probably slow). Is there a better way?</p>

You will need to use
Windows Image Acquisition (WIA)
to integrate a webcam with your application.  There are plenty examples of this readily available. Here is a
C# Webcam User Control
with source.
Here are some more articles and blog posts from people looking to solve the same problem you are:
MSDN Coding4Fun: Look at me! Windows Image Acquisition
CodeProject: WIA Scripting and .NET
CodeProject: WebCam Fast Image Capture Service using WIA
clausn.dk: Webcam control from C# and WIA