Is there a web service providing orbital data of solar bodies (planets, comets, asteroids, dwarf planets)?
Not quite the webservice, but there is a JavaScript library Orb2.js with examples including X-Y(-Z) in ecliptic coordinates. So maybe you can compute it directly, not relying on some external service:

http://www.lizard-tail.com/isana/lab/orbjs/

As far as I do not use it actively, it just a pointer to site to explore, not a definitive solution for you.