You may be able to get the dialog box (assuming that's what your screenshot represents) by using the UIAutomation module.
Once you've found the dialog box you can use the functions in the module to click on the "Cancel" button to dismiss the dialog box (if that's your intention) or to fill the text box and then click "Continue".
But that doesn't get you to an answer to the "find the process" part. For that you'd have to find every window and then check each to see if it has an open dialog box.