The claim here is that zoos and aquariums do a lot of good:
And therefore it would be another "crazy vegan" argument to be against them.
Zoos, as with anything, exist on a continuum. On the one end are possible non-profit animal sanctuaries that are established enough to sell plushies and t-shirts on the side to fund conservation and animal rescue efforts. On the other end is, for instance, the Waccatee Zoo, which is probably the most depressing image search of all time, and there's the most obvious one that gets all the documentaries - Seaworld. So you'll have to research this "zoo" you're thinking of heading to, and ask yourself what things it does.
One thing to consider is, for instance, that most Major-city zoos attempt to get a wide range of animals that the public will expect. This includes, for instance, elephants and other animals that normally roam long distances. If they obtained these animals by capturing them and removing them from the wild then that sort of is problematic for arguments that lean on "they are just rehabilitating and releasing animals". I would probably assume they are captured for most zoos, or kind of bred from some stock that these zoos share and buy off one another, and the number of animals that are rescued, rehabilitated, and re-released is probably negligible in comparison, but there doesn't seem to be a great way to find this information out.
Zoos also euthanize excess animals, and animals sometimes die or have to be put down as a result of accidents, whether it's the highly publicized Harambe or some shipping container of nameless penguins that arrive dead. Some initial research shows that neither scenario is that uncommon. So how they manage the animals is worth thinking about.
Zoos claim that they are based on education. This seems to be a dubious claim at best. In terms of information about animals, I would hesitate to assume that a kid got more out of visiting a zoo than reading a simple age-appropriate book. (In fact, as a kid I can name numerous things that I remember distinctly learning about sharks, penguins, and insects from books - I recall being fascinated by goblin sharks despite never seeing one in person. I can't tell you a single thing I recall learning about animals from visiting a zoo as a kid.) This is redoubled in the age of looking up documentaries instantly online.
The other claim is that they help in some way with the "connection" to the animals. But this is also somewhat dubious because you're equally having your kid form a bond with just indiscriminately caging animals before heading to the cafe area to get them a slice of pepperoni pizza (so much for teaching them to have a "connection" with animals). If you want to teach people how to have a connection with animals try just not eating them for a change, and if you want them to see animals in person - have them visit a proper animal sanctuary. So all in all I'm not quite sure that the education claim is truly cogent.
The other benefit that zoos mention is that they somehow promote conservation. So the claim here, more specifically, is that it's morally justified to imprison some animals if it means that we can conserve others of the same species. I wouldn't be doing my job if I didn't point out that most of these conservation efforts are to reverse the effects of carnists destroying habitats by things like farming food for livestock, poaching animals, and killing sharks and sea mammals as by-catch for fisheries.
The moral question thing is if it is indeed imposing suffering on animals to imprison them in cages - which seems like, again, has a sliding scale where on the one hand some animals, like stick insects, might have no awareness that they are imprisoned or might prefer the lack of predators, and on the other hand, some animals, like elephants and whales, that roam long distances that are confined to a tiny space. Now it should be argued that, for instance, there's nothing wrong with keeping a cat as an indoor cat as a means of protecting the animal from other animals and cars, just as we would say we have a responsibility to protect a child from things that might hurt it even though we're technically restricting freedoms as a result, but that's not carte blanche to keep your cat confined to a shoebox - so the type of enclosure is another sliding scale. However, it would seem as though most city zoos are attempting to get a good diversity of wildlife and therefore have no problem with the more problematic enclosures for roaming animals. So on a case-by-case basis, you'll have to decide whether you think the institution imposes suffering on its animals by caging them, and if so, that it justifies conservation efforts. We don't accept arguments about imposing suffering on some humans so that others might live a better life, so it doesn't seem like this argument would hold water.
The other question that we really need to ask ourselves is if this is indeed the best way to accomplish this. You'd be better off paying to see a nature documentary in an Imax theater that donates the proceeds to conservation, for instance, or donate that money directly yourself rather than paying a middleman.
Are zoos vegan-compatible? Animal sanctuaries show that this is not impossible in principle. But given all the issues with zoos where the positives seem to come up short and the negatives seem whitewashed, it's not some far-fetched crazy vegan argument that they should, at least in some part, be avoided if one doesn't want to contribute financially to animal suffering. So it's quite a reasonable stance to be against zoos for ethical reasons.