This sounds like the stuff of urban legends. No whale sperm, nor any whale product, is used in lip balm. Nor is any other creature's sperm for that matter. In the past, and to a small extent still today, ambergris, a waxy substance found in the intestines of sperm whales, was and is used in the production of certain perfumes. This may be the origin of the lip balm myth. However, the word "sperm" in sperm whale comes from the word spermaceti, an organ found in the whale's head.
Lip gloss can be made from a number of ingredients. Many are petroleum-based. Some use lanolin, carnauba wax, and other waxes. Many include natural ingredients such as coconut oil or sunflower oil. The only animal products that are used in lip balm that I am aware of come from bees. Beeswax is a common ingredient in natural lip balm. Some lip balms, like the brand popularized by Kourtney Kardashian, even include small amounts of bee venom, which has the effect of making lips swell up slightly for a plumper look.