If the shipping address you include in the query (along with the variant) is within the local delivery radius for the local delivery-enabled fulfillment location, the local delivery rate should pop up in the response output. Hope this helps!
This is for local PICKUP, not delivery. I tried this anyway, but it still did not surface the local pickup. I have a complete hack around your API that I use to supply customers with local pickup functionality for API generated orders, so I know it CAN work, it just doesn’t via the API.