Is it possible to automate order creation and checkout on my website?

My business utilizes an automation from Zapier that creates an order. Right now the order comes in as unpaid and an email is sent directly to the customer so they can view their order.

What I’d like to do, is hold the customer on my site, on a page that says “calculating price” while Zapier creates the order in my shopify. After the order comes into shopify a button appears at the bottom of the “calculating price” page, that the customer clicks and takes them to their order page where they can checkout utilizing a third-party subscription processer.

Is this possible? How can I achieve this?

Thank you community for your expertise.
Derrick