We have seen this before, I would check the Arcmap remoting settings for the machines that aren't working. To enable the button we check if the user is authorized( configured in rx web) and then if responder is licensed. You might also check the event log on the failing machines to see if there is any issues are being logged. Note this command is also available from explorer.
Shortcut to the solution: Log into Responder web as your administrator. On the Administration tab go into roles. Select the appropriate role for the user who should have access to this tool. Edit. Under Available Tasks, add task 'Reprocess Line Display'.
If that task does not show up in available tasks create it on the task tab and add the operation 'LINE_DISPLAY_REPROCESS_ALL'