Not sure if you're looking for a vegan cow's milk, but it doesn't exist. Cow's milk isn't vegan, no matter how the animals used to produce them were raised, because animal products aren't vegan. They are ovo-lacto vegetarian, but not vegan. Plant-based beverages, like soy milk, usually are vegan, though. Check the ingredient label to be sure.
At 16 months old, he should be able to have full-fat soy milk. Just make sure to get one that is fortified with calcium, vitamin D2 (vitamin D3 isn't vegan), and vitamin B12. Silk, Soy Dream, 8th Continent, Vitasoy, Edensoy, and Westsoy are some of the brands that you may find (there are others and many grocery store chains have their own in house brands of soy milk, too). They sell soy milk in cartons in the refrigerated section near the regular milk, they sell it in aseptic packages on the shelf, and they sell soy milk powder.
Steer clear from the low-fat or fat-free soy milk until he's 2 or 3 at least, though. Young children need the extra fat in their diet for eye and brain development and to help keep growing as they're so active at that age!
Soy is a common allergen though, so be sure to be careful when first giving it to him. Don't introduce any other new foods to him for probably the first week when you give it to him (so you can more easily rule out if he's allergic to soy or not). Offer small quantities at first and go from there. Don't go overboard with soy products, though. It's too easy to become dependent on them. For example, don't give him 3+ bottles/sip cups of soy milk in a day, some tofu with his lunch, edamame for a snack, and then a soy-based veggie burger at dinner. Like with anything, moderation is a good thing.