Is it possible for an approver to make the payment when they approve an order?

You can set two payment methods one would be pay on terms (Misc) and another one would be an integrated payment method. At the checkout page add an instruction informing the users to use the “misc” payment method so that they do not have to use their cards and once approver gets an email to approve this order he can select integrated payment option at the checkout page to make the payment.
For more information on setting up the payment method please refer to the chapter Ch. 037 Setting Up Payment Methods for Sites