https://ecode360.com/12010564?highlight=chickens&searchId=11134960126198079#12010564
SOURCE
[Emphasis Mine]
§ 348-5.24Animals.
[Amended 2-9-1982 by Ord. No. 2068; 9-25-1991 by Ord. No. 2859-91; 9-24-1996 by Ord. No. 3196-96; 12-26-2017 by Ord. No. 4569-17]
Animals shall be a permitted use in any residential zone, subject to the following:
A.
Adherence to the minimum health standards established and administered under the provisions of Chapters
165 and
516 of the Code of the Township of Toms River.
B.
*Structures used for the sheltering of horses shall not be located closer than 50 feet to any property line nor closer than 150 feet to any residence on another lot.
C.
Animals shall not be kept for commercial use.
D.
The keeping of poultry or livestock (other than horses and not more than six chickens, but excluding roosters) for personal use shall only be permitted in the Rural Residential Zone provided that all buildings and structures utilized for housing such livestock or poultry shall be set back in accordance with § 348-10.5E.
[cropped]
(6)
Minimum rear yard setback for principal and accessory buildings: 30 feet, except:
[...]
(c)
Building for the shelter of fewer than 100 poultry or fewer than two head of livestock: 50 feet.
(7)
Minimum side yard setback for principal and accessory building: 20 feet, except:
[...]
(c)
Building for the shelter of less than 100 poultry or less than two head of livestock: 50 feet.
So yes, you have multiple sources of potential recourse. First, they have to be zoned appropriately. "Small Backyards" and "Rural Residential" or "Rural" Zoning are not typically found in the same sentence. Second, even if they are appropriately zoned, the building ordinances demand that accessory buildings in those zones be located a minimum of 30 foot from the back property line and 20 foot from the side property line - except [for your purposes] in the case of small chicken coops, which need to be 50 foot from each line, respectively.
Chat with your neighbor. Direct them to BYC, we can help with managing the smell by differing coop practice - likely deep litter. Its low effort, low cost, and doesn't demand much of the neighbor. If they aren't amenable to it, you have the code enforcement route.