Is it possible to automatically pull random "tags" from a long string of text?
Of course it's possible, you pretty much described the algorithm to test, and it doesn't seem to contain any obviously non-computable steps:

Split the message into words
Filter out the common words
Sort the words by length
Pick the top ten and present them as tags

Not sure what you mean by "scalable level", this sounds client-side to me. Unless the messages are very long, i.e. not typed in by a human, I don't think there will be any problems just doing it.
I'm thinking if a user submits a message and they click a 'suggest tags' button, their message would be analyzed and a form field populated wIthaca random words from their post.
Is it possible to do this on a scalable level? Would JavaScript be able to handle it or better to Ajax back to python?
I'm thinking certain common words would be excluded (a, the, and, etc) and maybe the 10 longest words or just random not common words would be added to a form field like "tag1, tag2, tag3"