Good question! We know that since Asher visits a fish hatchery, most of the fish in the community are raised. More than likely they are actually genetically engineered. We know that there is at least a certain amount of genetic engineering happening with people, so it makes sense that it would happen with fish. They want everyone to have uniform food, and food is supplied for everyone. We know that there is a river, but I don’t think it would occur to anyone to fish from it. They wouldn’t need it to eat, and there is little to no leisure time in the community.
In The Giver, Jonas's community was based on a Utopian ideal, meaning that it was supposedly perfect. According to this society perfection is the absence of memory as well as everyone and thing having an assigned place and purpose. If you remember, there were people who worked at the fish hatchery in order to produce food for the community. The reasons people fish in today's society they are for recreation and to produce food. Recreation and idleness are not characteristics of this community so it is safe to assume that fishing would not be allowed based on that tenant of the community.
Also, since someone of the community has been assigned to produce fish for the community to eat, a person who fishes would be doing a job that has already been assigned to a community member. In doing this, them fishing for food becomes redundant since a community member has already been assigned to do it and the person would probably be looked upon in the community as being idle and forgetting their assigned responsibility. They would probably be disciplined. The assumption then would be that no, fishing would not be allowed in the community.