When using Lillie's Ribombee's "Inviting Wink" Ability to put Pokemon onto my opponent's Bench, are they allowed to evolve those Pokemon on their next turn since they just came into play?

Yes, as long as it's not the opponent's first turn of the game they can evolve those Pokemon if they are able to do so. It is a basic game rule that neither player can normally evolve any Pokemon on their first turn of the game, even if they are going second.

Relevant cards

SV9_67