Once they have at least 2k reputation a user can edit any post. Can a user who has less than 2k reputation be banned from suggesting edits if s/he has many edit suggestions rejected?
Yes, they can, per the Meta.SE thread which requested this feature and has since been marked as 'completed' meaning the feature is available. There's an automated system in place, I believe, though the specifics of that system are not public, so if they get enough edit rejects in a row it'll squish people.
That said, if there's a specific case where this may be necessary, consider reaching out to moderators (perhaps in chat, or through flags on a post using "Custom" reasons, and explain which user and which edit you're concerned about, but only once or twice, not many times) about the user, to ask us to look at their review actions and consider whether they're being overly disruptive or not.