I copied this off of a veterinary website, it's for dogs and cats, but it should be the same for goats. The only worry I'd have with sunscreen is that it might be toxic if they lick it off, but apparently there on sunscreens for pets...
Sunscreen, or sunblock, can be used on cats and dogs. Pets with light skin and a short or thin hair coat are particularly prone to sunburn, skin cancer, and other solar-induced skin diseases (solar dermatitis, actinic keratosis, actinic dermatitis). If your dog’s coat is shaved during the summer for cooling, sunscreen may also be helpful. Additionally, pets who have suffered hair loss from allergies, surgery, or cancer radiation can benefit from sunscreen.
In pets, sunburn can appear as red skin or hair loss. The most common sites for sunburn in cats and dogs are the bridge of the nose, ear tips, skin surrounding the lips, and any other area where skin pigmentation is low. The skin on the groin, inside legs, and abdomen can also need sunscreen because hair is very thin there and UV light can reflect off of concrete surfaces to affect those areas of skin. Also, dogs who like to expose their belly to the sun may need sunscreen.
The best way to minimize UV exposure in pets is to avoid the sun between 10 a.m. and 3 p.m. by keeping the pet inside. If that is not possible, sunscreen can be applied to protect the skin.
Select a sunscreen that is fragrance-free, non-staining, and contains UVA and UVB barriers similar to SPF 15 or SPF 30 for humans.
Some sunscreen products have been specifically created for pets. If you intend to use a sunscreen on a cat, be make certain that the product labeling specifically states that it is appropriate and safe for cats. Although some baby sunscreens may be safe for pets, avoid human sunscreens that have ingestion warnings because these products contain ingredients that can be toxic if licked by a dog or cat. No matter what sunscreen product you choose, remember to apply liberally and then re-apply regularly during sun exposure.
Additionally, there are UV protective clothing or sun suits for pets. These products may be helpful for protecting dogs who enjoy sunbathing on their backs and exposing the skin on their abdomen to the sun.