Forty Rod - I'd use canvas rather than deerskin unless you plan on using brain tan - regular tanned deer is a bear to bead on. Of course brain tan is expensive and just like regular buckskin will stretch, that's one of the advantages to canvas is doesn't stretch or at least not much. It will make the strips a bit stiffer, but then beaded strips are fairly stiff to begin with.
Like most Indians I don't use anything to hold the buckskin/cloth. I do know a few native bead workers who do use embroidery hoops etc, but only a very few.
If I were going to stretch a strip I'd just make up a piece using some 1" x 4" or such with two uprights at each end kind of like a bead loom - just stretch the canvas from end to end and tack down at each end. As I noted above leave enough extra around the beaded area to turn under and sew to the face from the backside - this will keep the edges from fraying. If you turned the edges along the long side under and tacked them down before attaching to the frame it would also help the material to keep from stretching/distorting as you bead.
BTW - what method of beadwork are you planning on using? Lane )what used to be called lazy stitch before the PC cops changed it) , appliqué, or
Lane is by far the quickest/easiest - if using lane I would start with a center lane and then work to the sides, alternating each lane so that the tension of the beadwork would equalize as you go along......Of course lane beaded strips were used mainly by the plains tribes such as the Lakota, Cheyenne, Kiowa and such, whereas the Mountain tribes such as the Blackfoot, Absaroka, Nez Perce, Shoshone, etc.used appliqué almost exclusively (the Absaroka aka Crow did often use a mix - a long center panel of appliqué with the beads going sideways and then bordered with lane stitch.
Hope this helps.....