There are two ways to make sure your customer receives an emailed invoice for their orders.
Invoice Receipt Method:
One way is a simple setting that exists on each company profile page, which allows you to choose a single contact for the company who should receive an invoice as soon as each order ships.
You'll find it by going to the company and then opening their edit page (the same place you'd go to edit their settings, permissions, addresses, etc)
Scroll to "Invoice Receipt Method", choose "Email" and then the person who should be the recipient of this company's invoices. This will send them an invoice each time an order hits shipped status.
The second option is Triggers
If you're looking for more flexibility (i.e. you want to send the invoice to more people, or at a status other than shipped status), you'll want to use triggers!
You can set up a trigger to fire at any status, choose the recipients, and attach the invoice.