In general, service animals must be permitted to accompany their handlers (the people that they assist) wherever those people are allowed, subject to reasonable exceptions.
This document from the U.S. DEPARTMENT OF HOUSING AND URBAN DEVELOPMENT on "Service Animals and Assistance Animals for People with Disabilities in Housing and
HUD-Funded Programs](https://portal.hud.gov/hudportal/documents/huddoc?id=servanimals_ntcfheo2013-01.pdf) discusses the rules which apply in housing situations. It says, after defining 'service animal" and "assistance animal, that:
If the animal meets the test for
"service animal," the animal must be permitted to accompany the individual with a disability to all areas of the facility where persons are normally allowed to go, unless
(1) the animal is out of control and its handler does not take effective action to control it;
(2) the animal is not housebroken (i.e., trained so that, absent illness or accident, the animal controls its waste elimination); or
(3) the animal poses a direct threat to the health or safety of others that cannot be
eliminated or reduced to an acceptable level by a reasonable modification to other policies, practices and procedures.
This would seem to preclude a requirement for a leash, provided that the animal is 'under control"
Guide dogs for the blind usually have a harness with a grip or handle for the person being aided to grab, rather than a leash. Other kinds of service animal may have other sorts of gear.
This site collects and summarizes the NJ legal positions on "Assistance Animal/Guide Dog Laws" The only mention of a leash is in 18A:46-13.3, which is specific to Schools where students need such assistance, and it says:
c. The service animal shall be under a handler's control at all times by use of a leash, tether, voice control, signal, or other suitable means. The school shall not be responsible or liable for the care or supervision of the service animal.
So even that provision does not seem to suggest that a leash is required if other means of control are sufficient.
It would seem that the local leash law is preempted by the federal housing regulations as far as a leash requirement is concerned.
However, I am not a lawyer. If the apartment management attempts to insist on enforcing a leash requirement, you would do well to consult a local lawyer knowledgeable about disability law and housing.