Is it possible to change the responsible party information on the EIN?

I am starting a nonprofit (military family readiness group) that will have completely new board members every 2 years, if not sooner. In other words, I will have nothing to do with this organization after I leave. I do not want someone having access to my social security number once I am gone. Is it possible to change to the responsible party information? If someone looks up the EIN, is my social security number going to be visible to them? Thanks!

Its not that your SSN is public if you're a "responsible party", no. It will not be visible. But its the IRS that would come to you knocking when problems arise years after your departure because they're not aware of any changes.

That is exactly why the IRS has announced new regulations, to come into effect next year, that would require periodic updates of the EIN information. So not only will you able to update the IRS about the change - you'll be required to.