A mi me gustaria poder tener la opción de yo en el calendario

A mi me gustaria poder tener la opción de yo en el calendario, marcar días aleatorios en los que se realizará la actividad, y que las persones marque en que dia realizaran la actividad.

I'm not sure I understood your question correctly. Do you want to your form users to check the dates that are still available?

You can try the Date Reservation Widget, this would let your users reserve the dates.

The widget displays a calendar where respondents can select a dates. There is an option to limit the number of days that can be selected. You also have the option to allow/disallow the selection of today, weekends, and past dates.

Please give it a try and let us know if this fits your requirement.