Ok, so I ran a quick search for you and here's three I found that roughly fit what you're looking sure. I'm not sure what your budget is so I listed a few for $30 and under. Needless to say, if you have more to spend, you have more options.
Here's one for $20 with a zipper. It is a faux leather, though, so it might still appeal to your dog.
this one is about $26 and seems to fit all your requirements. Another faux leather.
this wallet from Herschel for $30 It has a snap coin pocket rather than a zipper, but the effect is the same.
I can't attest to the quality of any of these, but they seem to roughly fit your request. Btw I found all of these by googling "men's wallet coin pocket" and variations thereof. Amazon, Zappos, and eBay are all good places to find lots of different styles at reasonable prices. Best of luck.
Bonus: if anyone else stumbles upon this post looking for a slim wallet, and you have a little more cash to spend I love my Bellroy wallet. Had it for about two years and it even fits in the tiny "pockets" of lady jeans. Lady jeans!!!