
Caracals, as pets, are illegal in New York state and New York City. I discuss the law below.
New York state
The following law applies if you are considering having a caracal as a pet and you live in New York State:
New York Consolidated Laws, Environmental Conservation Law – ENV § 11-0512. Possession, sale, barter, transfer, exchange and import of wild animals as pets prohibited. Here is a screenshot:

This law states that it is prohibited for any person to possess any wild animal for use as a pet in New York state. There are exceptions. There are two types of exception. Firstly, if you had a caracal as a pet before the law came into effect you can keep it under certain strict conditions. I can’t find out when the law was enacted. Secondly, organisations and institutions such as zoos and research facilities are allowed to have a caracal. There will be strict rules governing the keeping of the animal within these exceptions. You can read about those rules and exceptions by clicking on this link. Please note, however, that sometimes links to external websites break but I have no control over that.
There is a complete list of animals defined as being wild animals under New York legislation which you can read by clicking on the link below:
It is a very comprehensive list (under 11-0103) and it includes the caracal as expected. I cannot confirm conclusively that this list is up to date as at 2021. I suspect that it is. It includes, as expected, the caracal because essentially even small wild cats are dangerous animals. They can harm their owner and other people.

New York city
You would expect the rules on owning a caracal in New York City to be the same as those of New York state. On the official website of the city of New York, they have a page which tells us that animals which are not allowed to be kept as pets include the following cats: lions, tigers, leopards, jaguars, pumas, panthers, mountain lions, cheetahs, cougars, bobcats, lynxes, and other undomesticated cats. The caracal is not mentioned. However, they do state “and other undomesticated cats”. It seems to me that the caracal is an inadvertent omission but is caught within the law because it is an undomesticated cat.
The conclusion, therefore, is that is not legal to have a caracal as a pet in New York city. You can see the page that I’ve referred to by clicking on this link. Once again the sorts of external links can break so it doesn’t go to the webpage I’m sorry but I can’t control that. I have provided a screenshot as a backup:

If you have any questions please ask in a comment. Thanks.
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