Is it possible currently to state that a riddle should only be attempted once per user
What would be the purpose of this?
Given the whole Stack Exchange philosophy of "vote for the post, not for the user", it shouldn't even matter who posted an answer. If you get multiple answers to your riddle, then why do you care whether they're by the same person or multiple different people? I don't get the motivation here.
How would it be enforced?
The only real way to 'enforce' such a rule would be to expect moderators to delete a second answer posted by the same user. I don't think this is fair, either to the mods or to the user who might have posted a perfectly valid (maybe even correct) answer.
In fact, even this wouldn't really work as a means of enforcement. Someone who really wanted to post multiple answers to the same riddle could do either of the following, thus technically staying within the parameters of "at most one answer per user" while violating them in principle:
- delete their first answer (leaving it still visible to high-rep users) and post a second;
- edit their existing answer to mention the second solution instead of (or even as well as) the first.
In short, regardless of whether your suggestion is a sensible one, I don't think it's a practical one.
maybe add a specific tag to a question or something like that?
Definitely not. This type of tag would be what's called a meta tag, which are heavily frowned upon on Stack Exchange in general, including Puzzling. If you did want to make a requirement like this, it would have to be included in the text of the question.
Is this something that would be welcomed? Or frowned upon?
TL;DR: you could do it, but it might be considered odd and might not work well.
Other unusual requirements have been added to questions without issue. For example, this question stipulated that only low-rep users should answer it, which was largely respected (my answer was originally posted by accident, then - well, see the revision history :-) ) And this question stipulated that people should only solve it by hand - which indeed there's a whole tag no-computers for. So there is precedent for OPs adding their own conditions to a question and the community respecting that.