You don't have to do anything to legally inherit them but I would recommend getting an FID (Firearms ID) card to be able to purchase ammunition in NJ.

Here are the NJ statutes: 

13:54-1.13 Firearms passing to heirs or legatees 

(a) Notwithstanding the provisions of this subchapter concerning the acquisition of a firearm, a permit to purchase a 
handgun or a firearms purchaser identification card shall not be required of an heir or legatee for the passing of a firearm, 
upon the death of its owner. A person so acquiring ownership may retain the firearm if he or she meets the requirements 
of N.J.A.C. 13:54-1.5 and 1.6. 
(b) If an heir or legatee is not qualified to acquire a firearm, he or she may retain ownership of the firearm for the 
purpose of sale for a period of 180 days, which period may be extended by the chief of police or the Superintendent. 
During such period the firearm must be placed in the custody of the chief of police or Superintendent. 
(c) In the case of assault firearms or machine guns disposition shall be in accordance with N.J.A.C. 13:54-5..


13:54-1.5 Prerequisites for a permit to purchase handgun or a firearms purchaser identification card 
(a) Every person issued a permit to purchase a handgun, or a firearms purchaser identification card, shall be 18 years 
of age and of good repute in the community in which he lives. 
(b) A permit to purchase a handgun, or a firearm purchaser identification card, shall not be issued to any person: 
1. Who has been convicted of any crime; 
2. Who is drug dependent as defined by N.J.S.A. 24:21-2; 
3. Who has been or is then confined for a mental disorder in any institution; 
4. Who is a habitual drunkard or an alcoholic; 
5. Who suffers from a physical defect or disease which would make it unsafe for him or her to handle firearms; 
6. Who knowingly falsifies any information on the application form for a permit to purchase a handgun or firearms 
purchaser identification card; 
7. Who refuses to waive statutory or other rights of confidentiality relating to institutional confinement; or 
8. Where the issuance would otherwise not be in the interest of the public health, safety and welfare. 
13:54-1.6 Exception for physical disability, mental disorder or alcoholism 
A permit or identification card may be issued to a person who had previously suffered from a physical defect or disease, 
or mental disorder, or was an alcoholic if the applicant provides a certificate of a medical doctor or psychiatrist licensed 
in New Jersey, or other satisfactory proof that he or she is no longer suffering from that particular disability in such a 
manner that it would interfere with or handicap him or her in the handling of firearms.