Send invoice to customer to pay on website

Trying to figure out a solution for our specific needs.

We do website design, and want to send an "invoice link" to a customer to pay for services without the customer needing to add items to a cart themselves.
For example:

1. Site Admin adds multiple "1 hour" billable "items" to an order
2. send the customer an email to login.
3. Their "cart" indicates the amount due and they "checkout" to make payment.

Is this possible to do with Hikashop?

Yes.
Since 1.5.4, you can create new orders in the back end. You can also change its status and activate the user notification to send an email about his order to the user.
By default, there is no link to pay the order in that email, but you could give a link to the order listing of the user where he would see a 'pay now' button for his order: mywebsite.com/index.php?option=com_hikashop&ctrl=order

Yes this seems to work partially.

The problem is there doesn't seem to be a "check out" link when orders are created from the back end. I see the list of orders for a "customer" but no "check out' link? Something I am missing?

Ah yes, you need to turn on the option "Payment of orders not paid immediately" in the Checkout tab of the configuration for that.