ClickFunnels integrates with Authorize.net via Infusionsoft. Here is how it works. Infusionsoft is connected to your account through the Payment Integrations in My Account > Settings (inside of ClickFunnels). Then a product is created inside Infusionsoft. After that, you go to your Order Page inside of ClickFunnels, click on "Add Product" and then choose Infusionsoft as the billing integration. After that, you select the product from Infusionsoft in the drop down list.
Set the rest of the product up as needed, and then go to the E-mail Integrations tab. In there you can add Infusionsoft again and choose to run it through a Web Form into another campaign that adds various tags, etc or an Action Set.