By default date.toISOString returns UTC time instead of local time. So you'll need to calculate the timezone offset first.
//Get the local date in ISO format
var date = new Date();
date.setMinutes(date.getMinutes() - date.getTimezoneOffset());
var datestr = date.toISOString().substring(0, 10);
//Set the field value
var field = document.querySelector('#date');
field.value = datestr;
If it's a datetime field you're modifying (as opposed to just the date) don't forget to add the time T00:00, or change the substring to 16 characters for example:
//Get the local date and time in ISO format
var date = new Date();
date.setMinutes(date.getMinutes() - date.getTimezoneOffset());
var datestr = date.toISOString().substring(0, 16);
//Set the field value
var field = document.querySelector('#datetime');
field.value = datestr;