Are there shops that clean stuffed animals and teddy bears? Like dry cleaners? Or do dry cleaners normally do that?
You can probably wash them in your own washing machine, if you have one. If not, you can always bring it by a dry cleaner's and ask.
Just sleep with it as it is, a little dirt never hurt anyone. Remember the story of the Velveteen Rabbit and rejoice!
They make a cleaning solution for teddy bears and it works very well. Go into any bear and doll shop and they probably carry it. If not you can find it on Google. I used to be president of the world's largest teddy bear collectors' club so I know these things. If you have a commercially mass-produced stuffed toy it may very well be machine washable. Toys for kids are expected to get filthy. But if it's an artist-made collectible or even a high-ticket item by an outfit like Steiff or Hermann, with fine fabric like mohair, you have to treat it like the art object it is. Take a representative sample of your collection to the store where you look for bear bath soap and they should be able to tell you if it's machine washable, if the tag has disappeared.