If the screening is soft and pliable, you should be able to sew it on the machine. You will need miles of blanket or seam binding along all the edges since that will be all the reinforcing the edges have. I would take my sewing machine needle to the screen and see how easy it is to push through the screen “ thread” . You could set the machine stitch big enough that it sews in the holes between the threads but you know you will accidentally hit a few. Best to see if the needle will go through a screen thread well before trying it by machine.