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Well, it's pretty hard but try migrating Pokemon that has a move Sunny Day from Pokemon Fire red, Leaf green, Ruby, Sapphire and Emerald. Or just trade with a friend with that move. Forgive this answer. Though I've played this game 4 times with different Pokemon I forget where to find it.
Yes, the TM Sunny Day is in Pokemon Emerald. You can find it in a cave in the middle of the lake in Route 120 called Scorched Slab.
in the safari zone
TM11 Sunny Day can be found in the Safari Zone.
The Day Care in Pokemon Ruby is on the route West of Mauville City.