Taking a closer look into the query shared, I can confirm that the GraphQL Admin API channel(s) query was deprecated, as stated in our API documentation here.
The suggested replacement would be to use a publication(s) query,however this requires access to the scope read_publications available only to some apps installed on Shopify Plus stores. If you are building an app for a Plus store, please have them connect with their support contact to work through obtaining access.
To enquire about access for an app created within a partner account, please connect with our Partner Support team directly through the Partner Dashboard > Support menu.
We don’t have a Shopify Plus app yet, just a standard app. Did I understand you correctly, that this feature is only available to partners that have a Shopify Plus app, or would this feature be available to us if our client is on Shopify Plus?