There exist a number of non-animal-milk based options, however none of these are "better suited" than animal based formulas because humans have been giving their babies animal milk as a backup to breast milk for thousands of years (since both are dairy products) and the vast majority of research and product development has been used to improve these common alternatives over other ideas like putting a baby on a plant-based diet.
That said, some babies are actually allergic to dairy, so alternatives do exist. You'll be looking for anti-allergic baby formula. The most common alternative to dairy is to use soy-milk.
There are also some other alternatives posted in this answer.
Otherwise, consider talking to your pediatrician. They might be able to suggest other alternatives.
But don't be surprised if your pediatrician also recommends sticking with the regular dairy-based formula. It is the most natural and normal thing you can give an infant if you don't have enough breast milk.