How to add a calendar to a form?

Is there an existing form that would allow a user to access a calendar (weekly/monthly) and input date to be received by me? 

Yes, for example, you may add a date field to your form, this is the easiest way to do it: 

You may also search a template that fits your needs here: https://www.jotform.com/form-templates/ 

This guide will help you on how to use form templates: How-to-use-the-form-templates

We have also some widgets that may help you, you will find them here: https://widgets.jotform.com/category/pickers 

Follow the steps on this guide to add a widget to your form: How-to-Add-a-Widget-to-your-Form

Hope this helps.