How to use Commas When an Introductory Clause Precedes a Dependent/Subordinate Clause

There is nothing wrong with your use of commas. It follows normal guidelines, and should not have been questioned by your editor on those terms alone. (Many perfectly grammatical constructions use a number of commas.)

However, if there is an insistence on not using commas in such a way that they could be misinterpreted as parenthetical (there is some merit in reducing unnecessary parsing), then your sentence can rephrased by simply reversing the position of the conditional:

Accordingly, I want James to bring me some cheese if he goes to the store.

However, this is a matter of style rather than grammar.