Cooley: Since when did Sins have actual black holes?
There are two ways you can gain the ability to traverse wormholes: research, or a particular artifact.
For all factions, the wormhole travel research topic is located in their Civilian (or Advent/Vasari equivalent) research tree. However, the Vasari get their wormhole tech at Tier 4, where the TEC and Advent get this tech at Tier 5 (which makes sense, since the Vasari are more or less masters of Phase Space).
If you happen to find the Jump Field Generator artifact (it appears at the bottom right of the artifacts screen if you're a non-Entrenchment user) On top of reducing antimatter lost in phase jumps by 30%, it will grant your ships the ability to travel through wormholes, whether you've researched the wormhole tech or not. If you lose the artifact and you haven't performed the wormhole research, your ships lose the ability to travel through them (which has dire implications for ships stuck in an area where the only way out is through a wormhole).