There is no such behavior out-of-the-box.
You would need to customize the webpart to use some javascript modal windows instead of new pop-up windows.
You can find the webpart files in:
\CMSWebParts\Chat\ChatSupportRequest.ascx
\CMSWebParts\Chat\ChatSupportRequest.ascx.cs
\CMSWebParts\Chat\ChatSupportRequest_files\ChatSupportRequest.js
Dave