NABOWLA, Tasmania—Starting weeks before Christmas, Robert Ravens had to ration sales of teddy bears to people visiting his lavender farm in this remote corner of Australia to one per customer. Demand was so fierce that as soon as each furry purple Bobbie Bear shell was stuffed with a mixture of lavender and wheat, it was rushed straight to the gift store.

Mr. Ravens had already stopped shipping overseas and interstate. He had stopped taking Internet orders. His small staff at Bridestowe Lavender Estate couldn't keep up with...