I can't find a statement about the axiom of regularity anywhere in treatments of ETCS. Perhaps this is due to the unfortunate clash of terminology with 'foundations'.
As Noah suggests, the axiom of regularity doesn't make sense in the language of ETCS. What we can say is:
- To construct a model of ETCS+R from a model of ZFC, one doesn't need regularity.
- From a model of ETCS+R, one can construct a model of ZFC with regularity, and also separately models of ZFC with regularity replaced by various anti-foundation axioms.
The composite ETCS+R -> ZFC -> ETCS+R is the identity, as are the ill-founded versions. And the composite ZFC -> ETCS+R -> ZFC is also the identity, as are ill-founded versions whose anti-foundation axiom is sufficiently strong to characterize possible set-membership diagrams structurally.
Among other places, more details can be found in this paper of mine.