I get my supplies from www.micro-tools.com. In the left hand column there is a page for 'Rivet guns.' Take a look at the sizes of the rivets to see if they meet your needs. Since this a watch maker tool/parts supplier they ought to have some tiny rivets.
On the home page look under the header Watch Repair Tools. I see they have both US/EU departments. I found the rivets in the US section.