How can I enter "does not apply" via API or my eCommerce integration?

Via the API
We've added the param 'upc_does_not_apply'. Setting to true will apply the setting.

Via Shopify
A new custom tag called 'reverbsync-upc-does-not-apply:on' can be added to your Shopify Products.

Via Bigcommerce
A new custom field called 'reverb_upc_does_not_apply' should be set to value 'on'.