Will it ever rain on the island?
After about a month of Island trips I'm pretty confident it never rains on the island, even when it's raining at your town.
Instead, to catch a snail you'll need to plant a hibiscus bush in your town (they sell starter plants at the island shop sometimes, they cost 5 medals and come in assorted colors) and wait for it to rain, and you'll have a chance to see a snail. They seem somewhat rare but I have caught one using this method.
Yes, it can rain on the island, and when it does, you can find snails:
EDIT: If you're looking for rain on the island, you might want to check it in the morning. For the last two weeks, I've visited the island each day in the late afternoon or evening, and haven't seen it rain once during that time. However, I've seen it rain on three separate morning visits, at just past 8:00AM.