You can run it without brackets, but it really isn't recommended. The brackets are an integral part of the mechanical retention system that prevents the card from coming loose from its slot and maybe damaging things.
So if you must go that sort of route, what I'd suggest instead is that you take the existing half height bracket, unscrew it from the card, straighten the screwtab so it is flat in line with the rest of the bracket. Now the bracket will "fit" in a full height slot. But don't stop there. Insert the bracket into the chassis, and on the screwtab, take a Sharpie and mark a place to drill a hole. You want to drill a hole in this thing, and then go get a nut and a screw to clamp it to the rib between the PCIe slots. This leaves you with a ghetto solution that still securely clamps the card in place almost as well as the original mechanism.
Bear in mind that I would *much* rather see you acquire the right bracket, but if you have to go with an alternative solution, running without brackets is a poor choice in an expensive NAS. Find a way to lock it in there.
