Shop with request payment

I have been asked to help a friend of a friend set up a shopping/payment system on their Wordpress site

It is a dance business and they want a store front so people can pay for a taster class, one off lesson or products and a way to accept payments for invoices. The invoices are managed on a separate piece of software and currently parents bring in cheques every term.

I played with paypal request payment a few years ago which would work but does anyone know of a shop front plugin that will also accept a request for payment. I have found a few stand alone options but it seems a waste to have two different systems.
The way I see it working is the parent is emailed an invoice and they then login and pay the invoice using the reference number.

Edit: I have just found “Manual order creation” by accident on the Shopify website after spending about 15 min going round and searching their site.