Tanning season is apparently July 16 to September 15, but since the Island is summer all year you can get it there. I don't tan myself, but I believe you have to leave your 3ds open btw.
I've gotten tan from tours before. Even if you're beetle hunting at night, you can do tours and tan since most of the tours are in the daytime, even when it's nighttime on the island. And that place is always summer... such a weird little magical island
I got a tan today from just being in my town. And I didn't have my 3DS open. I shut it with my character standing by the train station and I went out for an hour or two, and when I opened my DS again, my character was hella tan. xD
Also try Club Tortimer at night time. Alot of the islands are set at daytime. The person in my timezone arrives and stays there.
Think it's only 30 minutes per day for a tan.