The LRIGs came first.
In Season 2 episode 8 "This World is My Property" it is revealed that:
The first LRIGs, Tama and Iona (then known as Shiro and Kuro) were created by Mayu to begin the LRIG and Selector system. These LRIGs are suggested to have special powers, compared to normal LRIGs. Mayu created more original LRIGs past the first two. Whenever one of these LRIGs loses three battles, they are not simply returned to the system to find another Selector; instead, they disappear. Tama and Iona are the only remaining original LRIGs.
Quoted from the Selector WIXOSS wikia & confirmed from viewing the episode in question.
Also, from another Wikia entry:
Mayu created two imaginary friends, Shiro and Kuro, in order to play the game with them, then proceeded to conjure up the idea of the Selector Games in her mind strictly for her own amusement, creating yet more imaginary friends which served as the first Selectors and LRIGs. Eventually, Mayu's make-belief games became so realistic that the imaginary friends took a life of their own. Mayu proceeded to send them out to WIXOSS players in the outside world, effectively "infecting" the game with her own, selfish desires.