You're unlikely to find spares listed for a small, cheap player like that. Your best bet is to see if you can find a push latch of the right size, shape and fixing on any of the tat bazaars like eBay, Amazon, Banggood, AliExpress, Dealextreme, etc. It's a lot of legwork.
Why not just rip your DVDs to mp4 videos, pop a large memory card in your phone and watch on that? If it's for your kids, give them an old phone with apps you don't want them to use, hidden away.