How do you get the Season 2016 - Solo Bronze?
It was a ranked reward. Riot's description in the client reads: “Earned by achieving Bronze in the 2016 Ranked Season in Solo.”
Is the Season 2016 - Solo Bronze still available?
No. The Season 2016 - Solo Bronze was a ranked reward: to get it you had to reach Bronze in Solo/Duo during Season 6 (2016). Riot does not give out past ranked rewards again, so it cannot be earned anymore.
