As Japan moves into the Reiwa (令和) era, I was wondering if any traditional greetings exist to congratulate someone on the new Era, Emperor or Year of the Japanese calendar?
No, there's no such greeting. In the last 150 years or so, a new era has been associated with the previous Emperor's demise, so it's not a happy event in the first place. This time is an exception, and people may be allowed to say something including おめでとうございます, but there is no fixed, traditional way of saying congratulations.
(Disclaimer: I'm writing this from the standpoint of an ordinary native speaker. Experts may know some rare phrase that was used long ago.)