The only problem here is not with neither ... nor per se but with the ambiguity that arises from using the modal to need with a negation. Consider the following:
- You need to do that!
- You don't need to do that!
- You need not to do that!
- You need not do that!
It's not at all logical that 4 expresses the same way of negating 1 as 2 does, and not the one from 3, to which it is structurally most similar. The reason for this inconsistency is that modals can still be negated the old-fashioned way, with not after the conjugated verb - resulting in the placement of not in the same position as when negating the infinitive that follows.
We are used to dealing with this difficulty correctly and efficiently so long as we are on familiar territory. But combining a modal with neither ... nor is relatively rare and the emerging convention here of binding the negation to the modal rather than the infinitives is illogical and perhaps not universally understood. Since the inherent negation in neither ... nor is part of a bracket that binds the two infinitives to store and to access together, it seems essentially unavoidable that some people will read the negation as negating these two infinitives (individually) rather than the modal to need. This results in the following reading of your sentence:
It is necessary that he neither stores the datasets locally nor accesses the outsourced datasets each time they delegate the computation.
To prevent this unwanted reading while still using neither ... nor for brevity, you can rephrase as follows:
He neither needs to store the datasets locally nor [does he need] to access the outsourced datasets each time they delegate the computation.
The part in brackets is optional. It makes the intent even clearer, but also makes the sentence longer and more formal-sounding. A more colloquial alternative is as follows:
He needn't store the datasets locally nor access the outsourced datasets each time they delegate the computation.