If "No" answer, promte for reason

Is there a way for a answer box to appear for the person filling it out to put the reason why if they answer no to a multiple choice question?

You can achieve that by adding conditional logic on your form.

Please refer to this guide : Smart Forms Using Conditional Logic

Here's my example form, you can clone this form to find out how it works.

https://form.jotform.com/82875159725975

Here's a guide How to Clone an Existing Form from a URL

Let us know if you need any further assistance. We will be happy to help.