Since onions are 80%-90% water, the juice of a crisp, firm onion adds moisture and savoriness to a dry patty. All you have to do is coax the juice out of the onion into your patty.
Place raw onion slices on your veggie burger after you've flipped it on the first side. The onion will partially release some of its intrinsic juices as the other side sizzles.
Timing is key, so leave your patty in the pan long enough to soften your onion but not too long to overcook the other side. Otherwise, this will exacerbate the patty's dryness.
This onion hack is most useful for store-bought burger varieties with lower moisture and fat content that tend to lose their water-based moisture when heated.