Postdoc without PhD?

At this point in time, a Ph.D. is generally a non-negotiable requirement for a postdoc position. If you are already carrying out independent research, however, then it might be possible for you to obtain a Ph.D. in graduate school in a significantly shorter period of time than normal.

What are you hoping to get out of a postdoc? Typically postdocs offer a chance to enhance your research credentials and capabilities before moving on to a more permanent position either in academia or a serious research institution. If you do not have a PhD and wish to follow this career path, the best way to enhance your credentials and capabilities would be to pursue a PhD. A very capable PhD student can certainly do work of equal quality to postdoctoral researchers while also gaining a degree in the process. The only drawbacks to a PhD position compared to a postdoc seem to be somewhat lower pay and receiving slightly less individual credit for your work (in some fields anyway).

To more directly answer your question: Given enough funding flexibility, it's not unheard of for a professor to fill an advertised postdoctoral position with a very capable grad student. It would be very uncommon for a non-PhD, non-grad student to hold such a position.

As an aside, by definition you cannot hold a postdoctoral position without having a doctorate. The term literally means "after doctorate" and therefore requiress first gaining a doctorate. Any position you hold before obtaining a doctorate is by definition a pre-doctoral position.

You should certainly apply for grad school!

If your credentials are as you say, you will probably have the opportunity to go to one of the very best universities. Is there anyone whose work you have always admired? You now possibly have the chance to study under him or her. Take advantage of it!