Similar to requesting the zap signatures from the user the payment can be called with window.webln
WebLN is a set of specifications for Lightning apps and client providers to facilitate communication between web apps and users' Lightning nodes in a secure way. It provides a programmatic, permissioned interface for letting applications ask users to send payments, generate invoices to receive payments, and much more.
Similar to requesting the zap signatures from the user the payment can be called with window.webln
https://www.webln.guide/
This is a quick example on how it could look like. (edited on GitHub, so handle with care, but I hope it is a start)