How to show payment field based on user selection?

I have two options with a radio button and would like one of the options to open up additional details. For example, one option says "mail in payment" and the other option says "pay online." I want the pay online button to open up a paypal payment option so they can complete their online payment. I have had many customers choose to mail in payment but they are also filling out the paypal payment option. When they do that option without paying immediately online, their registration goes into the incomplete payments tab and I never get an email when they register. Is this available?

I have checked your form and seems like you were already able to apply the functionality that you need, I can see when the option " No charge: I will mail in my payment/deposit" is selected in the field "Choose your payment option" the payment field does not appear, here is a screen-cast about my test: 

Please do note that when the payment field is hidden your users will not be redirected to the PayPal payment page, no matter if they have already selected one option in the payment field. 

Here  is also a guide that will provide you with more details about show / hide fields based on other field's answers: How-to-Show-or-Hide-Fields-Base-on-User-s-Answer

You may also let us know if you need assistance, we will be glad to help. 

Thank you for the feedback. I tested the submit button and it redirected me to PayPal. Are you sure the users will not be directed there?
Thanks,
Jeremy
...

Yes, I have just tested your form and I was redirected to the Thank You page although I have selected some options in the payment section.

Here's a screen-cast about it: 

I have also checked and I can see you have not any hidden field in your form only the payment field hat you're hiding with conditions, if the payment field does not have an option selected it will not redirect to the PayPal payment page even if it is shown, I would suggest you to clear hidden fields this way you will avoid selections made on the payment field to stay selected when the option to mail the check is selected. 

To do it, click on the preferences icon: 

Go to the advanced tab and the show more button: 

There you will find the option to clear hidden fields: 

Please give it a try and let us know how it goes.