Have a look at the change mat I made for a friend's brand new baby
The product is called Iron on Vinyl & is made by Therm o Web. I ordered it online - EBay I think.
It is 43cm wide, but if you need it wider (as I did for this change mat) you just use multiple strips and overlap a little, iron on & the join disappears! It is easy to use, and doesn't bubble up or stick like mad - I had visions of that stick on plastic we used to use to cover files that would inevitably stick in the wrong place, or stick to itself, and you would end up in a big mess, uttering words your mother would not like to hear you mutter. BUT none of that happened. It is really easy to apply & irons on quickly.
I recommend that you keep the backing paper for re-ironing it after you sew - this will fill in the pin holes you have made!
I would also suggest you include this backing paper with the item, as the new owner of the change mat will need it to iron the mat after washing. Otherwise they will need to find very smooth fabric, but the paper is easy & quick!