Vegetable oil does not work well because it beads up on top of the water and does not spread across the surface. It also goes rancid.
Mosquito larva breathe air through a little tube while they float on the surface, so "oil" remedies are designed to cut off their air supply. You can try a horticultural spray oil. It only takes a few drops to form a microscopically thin sheen across the water. This is recommended for bird baths so it's pretty safe for the innocent bystanders in your yard.
You can also try mineral oil; Completely non-toxic. There are also non-chemical larvicides (that are mostly mineral oil) that are made specifically for this.
These pet-safe mosquito dunks contain Bacillus thuringiensis (Bti bacteria) which mosquito larva eat and die but it is completely harmless to everything else: pets, birds, wildlife, etc.
If your pond water is otherwise healthy, you can always try raising some fish. Goldfish, koi or, yes, even the aptly-named "mosquitofish" love the stuff.