How can I collect their credit card info after the order is placed?

There are two options:

1)  Add a product to your store called "Pay Invoice". See for an example. After the order is placed, have them use this to pay for the order.  The product uses a combination of a form and a pricing engine.  To create this follow these instructions:
a)  Steps 2 and 3 on this page are a pricing calculator.  Click here to download the spreadsheet used to create this pricing engine.  You would need to create a pricing calculator using the attached spreadsheet.
b)  Create the invoice product and assign the pricing engine you just created.  When you assign the pricing engine, make sure to mark that Q1 should allow arbitrary quantities.
c)  Step 1 on this page is a simple form.  To create a form, go to Preferences > Forms.  See this article for instructions:  Ch. 019. Forms  Once the form is created, assign it to the product.
2)  Call them and collect the information over the phone.