If the leadership in Salt Lake City have not given guidelines on a given topic, can a local unit make policy? For instance, old Primary guidelines precluded snacks for the Primary children, but the newest guidelines say nothing. My local Primary president thinks she can disallow snacks, I think if the leadership has not set a policy, a local unit cannot set policy either. Or if the Salt Lake City leaders have not spoken out on wearing denim in church, can a local unit outlaw wearing denim?
The local leaders (Bishop and Stake President) have the keys to preside over their areas. This includes the ability and authority to enact local policies to meet the needs of their stewardships. They can then delegate this authority to various auxiliary presidents so they can manage their stewardships.
So yes a Primary President can say no snacks in primary. However the ability to set such policy does not necessary make such a policy wise. For example the denim restriction might be good if you have people that are just being lazy or otherwise not putting forth their best. But for those to whom denim is their best or who have never be taught such a restriction is an unnecessary burden. A good leader should be seeking out the Lord’s guidance in such things.