Postback on TextBox loosing focus
If you just want to do a postback when TextBox looses focus (by pressing Enter, Tab or moving focus to another control), set AutoPostBack property of the TextBox to true. 
After writing something in the TextBox and making it loose focus, you'll get a postback. I suppose, however, that this can be raised not only on pressing Enter.
More information on MSDN: http://msdn.microsoft.com/en-us/library/system.web.ui.webcontrols.textbox.autopostback.aspx.
Postback by simulating submit Button click on pressing Enter
If you want a postback only on pressing enter, you can put add a Button on the page and put it with the TextBox in the same Panel control. Then you should set DefaultButton property for this Panel. This should invoke button click on pressing Enter when the TextBox (or any other control in the Panel) is focused. 
So, your code can look like this (the code hasn't been tested):
...
<asp:Panel ID="grouppingPanel" runat="server" DefaultButton="btnSubmitDomain">
    <asp:TextBox ID="txtDomain" runat="server" ToolTip="Network domain user is associated." />
    <asp:Label ID="lblRqdDom" runat="server" Text="required" CssClass="noshow" ></asp:Label>
    <asp:Button ID="btnSubmitDomain" runat="server" Text="GO" OnClick="btnSubmitDomain_Click" />
</asp:Panel>
...
You can find more information about it here: http://msdn.microsoft.com/en-us/library/system.web.ui.webcontrols.panel.defaultbutton.aspx.
If you have any issues applying any of those solutions or further questions, let me know.