I'm recently back from a trip to the UK which included a visit to the Victoria and Albert Museum East Storehouse (pictured above) near Stratford. Using their Order an Object service, you can arrange to view up to 5 objects from the museum collection that aren't ordinarily on display. I went to see some tablet weaving, naturally, but they have a vast array of other objects available to view. Out of all my trips behind-the-scenes, this was definitely the easiest to set up!
It's often possible to view objects that aren't on display at other museums, too. First, find the catalogue number of what you want to see, then email the curator with your request and your reasons. I find "I am an independent researcher with a special interest in tablet weaving" usually does the trick. You may occasionally get a reply that they don't have the facilities to host you or the object isn't available, but it's always worth asking. Generally curators are excited that you're excited about the object, especially if it's rarely viewed.
Be prepared to leave your bags in a locker or cubby before you view your objects and don't take in any food, drink or pens. You may be asked to sign a document stating what you can and can't do with any pictures you may take. You may also be required to wash your hands, remove any rings you're wearing, or wear gloves. My standard kit for object viewings is a pad of graph paper, several mechanical pencils, a tape measure or ruler (depending on the size of the objects), a protractor (to measure twining angles), and some textile record sheets that I've pre-filled with the information on the museum's website.
You can find the textile record sheets I use here: Link