Yes, staple gun and fabric. Not sure why the lack of creases is a factor. You get it set properly, staple it tightly, and it doesn't move.
I'm not a fan of vinyl as I stick to it, but it is waterproof. You can also get various other fabrics that are water proof or resistant. You might look at some of the outdoor upholstery fabrics. Make sure whatever you get is upholstery, it's heavier weight and will last longer.