What computer operating system is your Premiere Elements 12 running on?
I do not see Video Stabilizer Effect listed as Windows Only, so regardless of what computer operating system your Premiere Elements 12 is running on, you should have the Effect. Another thought is "Are you in the Expert or the Quick workspace".
In the Expert workspace ONLY, please check out fx Effects Tab (bottom of the interface)/Video Stabilizer/Stabilizer.
Please let us know if you found the Effect.
Thanks for the good news.
Just in case note, after you apply the Video Stabilizer effect to your video, when you go to render the Timeline content to get the best possible preview of what you are seeing in the Edit area monitor, do not be alarmed if this type of rendering time is longer than expected - can be dramatic longer than expected.
Please do not hesitate to ask questions or ask for clarification.
Your follow up was appreciated.