All ShopRites have many 'natural' products in most parts of the store, though you'll likely need to check product labels for such info. The stores are large with plenty of items from which to choose.
They have a few isles dedicated to "all natural" and "whole foods" as well as their new "Wholesome Pantry" line. They sell non-dairy milks, but I'm not sure what you mean by "without oils and nuts". Perhaps you can be more specific with regard to preference brands or items. Otherwise, preparing home meals with out those ingredients might be a more desirable way to prepare your meals. Are you asking for "ready to eat prepared meals" from their kitchen?
They do have an aisle dedicated to natural items
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