For every (finite simple) graph there exists a regular supergraph, even with the original graph as an induced subgraph.
Both versions (chronologically first the induced one, then the normal one with a reduced number of additional vertices needed) were proven by known graph theorists in I believe the sixties, but I fail to find the papers again and I would like you to help me find them.
The induced version of the problem is solved by Erdős and Kelly in "The minimal regular graph containing a given graph" (1963).
The other version is solved by Akiyama in "Regular graphs containing a given graph" (1983).