Are you sure it is actually an update from the manufacturer/OS creator? It may not be if it isn't working. Or you could just have no room on the tablet for the update.
First I would try checking the tablet for malware, viruses, bad apps, etc., to make sure that this 'update' notice you are getting is not actually something bad for your tablet.
Should you find nothing of that kind on the device, then try contacting the manufacturer and see if this is indeed a legitimate critical update. If it is, they would be able to tell you and they can also tell you how much room that you will need on the device to do the update. Then you can see if you have enough or need to remove things to get more room.