When they click "go"/"send" and send off the email, take them to a screen telling them its been submitted.
I was thinking maybe a, for lack of better word, "pop-up", telling them the request has been emailed and will be reviewed by staff as soon as possible. Then they click "ok" to close this message and are redirected back to Special:AdminPanel.
What @cook879 is describing can be achieved with the jQuery UI dialog component (ResourceLoader module jquery.ui.dialog). Example use can be seen on Special:CreateWiki, step #3, when the chosen wiki type is "private" and you attempt to proceed to step 4 but both viewing and editing restrictions are set to "everyone" (in which case it obviously wouldn't be a private wiki!). The code for this can be found at /extensions/CreateWiki/scripts/createwiki.js, function checkForErrorsPage3. As always, ping me if you need help with this -- ResourceLoader is somewhat of a messy system even from the viewpoint of an experienced developer!
So, "pop-up" box exists. There are still some issues with it, but they can go in a new ticket.