That functionality isn't exposed by the .NET WebBrowser control. Fortunately, that control is mostly a wrapper around the 'old' control. This means you can subscribe to the BeforeNavigate2 event you know and love(?) using something like the following (after adding a reference to SHDocVw to your project):
Dim ie = DirectCast(WebBrowser1.ActiveXInstance, SHDocVw.InternetExplorer)
AddHandler ie.BeforeNavigate2, AddressOf WebBrowser_BeforeNavigate2
...and do whatever you want to the PostData inside that event:
Private Sub WebBrowser_BeforeNavigate2(ByVal pDisp As Object, ByRef URL As Object, _
ByRef Flags As Object, ByRef TargetFrameName As Object, _
ByRef PostData As Object, ByRef Headers As Object, ByRef Cancel As Boolean)
Dim PostDataText = System.Text.Encoding.ASCII.GetString(PostData)
End Sub
One important caveat: the documentation for the WebBrowser.ActiveXInstance property states that "This API supports the .NET Framework infrastructure and is not intended to be used directly from your code.". In other words: your use of the property may break your app at any point in the future, for example when the Framework people decide to implement their own browser component, instead of wrapping the existing SHDocVw COM one.
So, you'll not want to put this code in anything you ship to a lot of people and/or anything that should remain working for many Framework versions to come...