User change order status after checkout

Hi,

It's totally possible yes.
You'll want to disable the "order creation notification" via the menu System>Emails in order to not notify the user at the end of the checkout.
You'll also want to go in the System>Payment methods to either disable the payment methods or set then to a user group that the user doesn't have access to, or have only one offline payment method (like collect on delivery for example) renamed to "payment will happen after review" or something like that.
You'll then want to go to the HikaShop configuration and remove the "payment" view from the checkout workflow so that no payment happens at the end of the checkout (or not if you want to use an offline payment method)
Finally, you'll want to go in the System>Order statuses to configure the order statuses so that the use can pay for the order when the order is in a specific status with the unpaid column.
Then, after an order is created by a customer, only you will be notified by email as admin, and you'll be able to review it, maybe change the payment method selected if necessary, change its status to the one enabled in the unpaid column, and notify the customer with the checkbox provided. That will then send to the customer an email with a payment link he will be able to click on to pay for the order.