Very few good zoos can afford to devote realistic sums of money towards defined programmes of research. But they can nonetheless encourage research in a number of ways. To begin with they could offer training to keepers in the scientific method, could encourage them to keep accurate records and observations of the behaviour, health, and growth of the animals in their charge, and could then offer them a medium in which to publish any significant findings. You never know what might prove to be useful. So record it all.
Good zoos share the information they gather.
Some employ their own full time veterinary staff, both to treat sick animals, and to study ways of maintaining animals in a healthy state. They support and solicit research from universities or research establishments nearby. Behavioural research is particularly suited to zoos. One way to measure how 'happy' zoo animals are is to study and compare their behaviour in the wild, and their behaviour in the zoo. If they are spending the same proportions of their time doing the same sort of things, then they are probably okay.