Can we collect the payments and have them go directly to our checking account?

Unfortunately that is not possible. You will need to use one of the payment integrations available in jotform. Our payment integrations simply allow payments through their services. So for example, when using Stripe payment integration to collect payments, all the payments are processed through Stripe. With this in mind, perhaps you can setup your bank account with your Stripe account so that the payments will go there.

Please see this guide for more info: 

Order Form Basics

Setting Up Your First Order Form

Regards.