There is nothing built into Store that allows you to limit access to products to certain members. Having said that there are a few ways you can get around this. A while back I wrote a blog post about using Simple Commerce for creating invoices. While doing this with Store will be a bit different you might be able to apply some of the template logic or ideas into your setup.
1.) You will need to use an add-on to send the email to the client. You can use something like MX Notify or PostMaster to send the email when you submit the order (I would probably combine this with a status parameter so you can edit/draft the invoice before it is sent).
2.) You would again need to use an add-on to have a due date automatically sent. Personally I set the due dates manually but something like AutoExpire should do the trick, just remember to allow expired entries in your channel entries tags on the front-end.
All that said, there is not an out of the box solution that will allow you to do something like this, but it very doable.
If you are still using Store v1 oppose to the newly released Store v2 you might be able to use Store: Quotations for your needs. I have never used it personally so I am not sure of all of it's features.