In reply to mattc:
There's quite a bit within reasonable distance:
https://www.thecrag.com/climbing/germany/maps#48.42,9.46,7.7,,auto
I've only climbed in the Donautal, which gives great limestone routes up to 120m. They're a mixture of sport and trad; about 1 1/2 hours drive from Stuttgart. The Schwäbische Alb is closer and also has many good possibilities; also limestone and shorter routes.
I think there's a UKC poster who is based there; can't remember his name though.
Edit: maybe MischaHY can help you, he's apparently based in Tübingen which isn't far from Stuttgart.
PPS the DAV Kletterhalle in Stuttgart is huge and very good; the German national team trains there regularly.
Post edited at 17:38