Hello Ann! As Simon said, it refers to both animals and vegetables. It is not ambiguous, and cannot mean "many animals and also an indeterminate amount of vegetables." It only means "many animals and many vegetables."

In order to get away from knowledge of farms, let's try "in my desk drawer there are many pens and pencils." I would expect this to mean both many pens and also many pencils. Why? Because if the speaker didn't mean this, they would have said something different, like: "In my desk drawer, there are many pens and a few pencils." "In my desk drawer, there are many pens and some pencils."

It means there are many animals and many vegetables. It's not ambiguous it only has one meaning - there are many animals and many vegetables on the farm.

It refers to both animals and vegetables

This is a fascinating question. I don't think this is a question of analyzing grammar, it's more of a question of how we understand real language in the real world. We understand it based on what we know about farms. As you have written it, I would expect it to refer to both animals and vegetables. It's not ambiguous. Here are two farms that do NOT have "many animals and vegetables:" Farm A has thirty pigs, fifty sheep, forty cows, and six tomato plants in the farmhouse garden. Farm B has two horses, twenty hectares of lettuce, twenty hectares of corn, and a twenty-hectare apple orchard. If a farm has "many animals" I would expect that to mean that if you counted the animals, it would be a large number. But I would also expect there to be several different KINDS of animal, to justify using the word "animal" instead of using the name of the particular animal. A farm that raises vegetables would always be a large number of individual plants, so I would expect "many vegetables" to mean "many different kinds of vegetable." When talking to children who experience the farm almost like a zoo, I would say "animals." In a more formal, adult, or professional context, I would use the uncountable noun "livestock" instead of "animals," and "crops" instead of "vegetables."