Anders from the Shopify Social Care team here. Thanks for reaching out today!
When you, as the account owner of a Shopify Store, sign up for our Shopify Payments, there is no individual processor account made. Each shop is a part of a single Shopify Payments system. Since you're applying for a California tax permit, instead, I can provide your shop ID number. This, along with an explanation to your local tax office, is often enough. However, this does mean you most likely need to speak with someone directly rather than applying online. But if there is a field to explain this in your online application, then great!
I've just sent a message to the email listed as the account owner of your store with further details. Speak soon!