stable homotopy - Are there universe-indexed spectra over simplicial sets?

Yes to both interpretations of your question. It is not clear to me where you want to put pointed simplicial sets.

One interpretation of your question is that you want to replace pointed topological spaces with pointed simplicial giving the notion of a spectrum as a functor from supspaces of U to pointed sSet. This is a very common thing to do, and in some circles in homotopy theory is the standard definition of spectrum. Often "space" is interpreted as meaning simplicial set. This is because of the standard Quillen equivalence between Top and sSet.

The other possible interpretation of your question is that you are trying to replace vector spaces with simplicial sets. This is also (essentially) something which has been done. It gives a model of spectra known as W-spaces. This is one of the standard diagram category models of spectra. See the following paper for a comparison:

Model categories of diagram spectra, by M. A. Mandell, J. P. May, S. Schwede, and B. Shipley