To intercept HTTP traffic using .NET, you need to create a web proxy and direct your WebBrowser Control to that proxy. There's an extensive discussion here:
How to create a simple proxy in C#?
Having said that, you will almost certainly find that the communication is encrypted. After all, you would not want someone else to be able to intercept which cards are in your hand just by sniffing packets on the wire.
Many online poker games allow you to download hand history using a well-defined mechanism after a hand completes. All of the poker analysis tools I'm familiar with use that method.
If for some reason you can't do that, consider instead doing a screen capture when the hands are shown. You can look at the screen at key positions where card values are shown and compare to the image of each of the 52 possible cards (or more likely just the area where the value and suit are shown) to determine which card is in each position.