Create a form within a monthly grid calendar

I am trying to publish a form with a monthly calendar on my website where users can place a check mark on any date when they are not available. I would also like all the unavailable dates to be exported to a spreadsheet, so that can see who is unavailable on any given date.  Is there a form or add-on or integration that will allow this type of workflow?

You can add DateTime filed to your form to allow your users to select date when they are not available.

You can also use widget like Date Picker. In case that you do not want to allow different users to select same date you can use Date Reservation widget. 

How to Add a Widget to your Form?

You can export your form data to Excel or integrate your form with Google Spreadsheet:

https://www.jotform.com/help/44-How-to-Export-Form-Data-to-Excel 

https://www.jotform.com/help/228-How-to-Integrate-Form-with-Google-Spreadsheet 

Hope this will help. Let us know if you need further assistance.