Is there a service that provides Battle.NET badges to include in web pages?
I believe they do not offer this.
Sites like SC2Ranks for example, scrape the battle.net public profiles in order to generate their statistics.
Your best bet would be to do that, or even check their API out to see if you can skip that step and take advantage of their work (of course, this is starcraft 2 specific, but there may be similar things for other games that I don't know about)
I think what you are looking for is this website.
It generates images like this:
