Sounds like an awesome idea to me! Go ahead, although you might like to put links in pointing the way to the next 'chapter' so people can follow the story along and not have to go searching for the next part. Double drabbles (200 words) are also allowed, so you can write longer chapters if you want.

I believe there's one author who's been doing something similar from the start, continuing her story with each new prompt.

I just post the prompts, how people use them is up to each writer to decide.

Have fun!