>"The same ones that produce the energy of the vacuum."

  in mainstream physics theories the vacuum has zero energy, and [feynman diagram loops representing creation and annihilation of particle pairs][1] are non existent without an incoming real particle traversing the vacuum.

>do virtual particles pop in and out of existence in the space just outside the nucleus?

A *nucleus* can be treated quantum mechanically with feynman diagrams as a real input line in a diagram , and thus loop corrections can be envisaged between the interactions of the nucleus with the lowest electron orbital field. So in this sense yes, virtual particle antiparticle loops are possible, except they are such higher order corrections that the powers of 1/137 that will enter in the calculations , make it a futile effort as a correction. 

For the total atom, the [Lamb shift][2] is the effect of such  vacuum loops modifying calculations.


  [1]: http://physics.stackexchange.com/q/185110
  [2]: https://en.wikipedia.org/wiki/Lamb_shift