When you see some dates are grayed out and not able to be selected, that means those dates are restricted for delivery (or pick-up in case of pick-up).

Please check the allowable delivery weekdays settings, have you enabled each day or some of days as allowable delivery day(s) by checking the check box regarding each of weekdays?
If you leave as unchecked for some weekdays then dates fall on those days will not be clickable on date-picker calendar.
Suppose you have checked only Monday and Wednesday as allowable delivery days, so all the dates fall on rest of weekdays will not be clickable on date-picker calendar, lets say 18th November,2016 is not clickable coz that dates falls on Friday and you did not check the checkbox for Friday as allowable delivery day.

Check you holiday settings if you have selected some dates as either casual holidays or national holidays, dates you declared as holidays are disable to select/pickup on date-picker calendar.

If you still can’t pick dates from date-picker please read this topic “Calendar issue for languages other than English

Hope it resolve your issue, if not please send us wp-admin and FTP login access along with your site url and please keep the plugin activated so that we can check what causing the issue for you.