Listen for the focus event on the element, then invoke the HTMLInputElement.showPicker() method:

document.getElementById("datePicker").addEventListener('focus', function(event) {
  event.target.showPicker();
});

<input type="date" id="datePicker" />

^{Note: Unfortunately this doesn't work in the snippet above due to the restrictions of calling showPicker() cross-origin combined with the cross-origin implications of the Stack Snippets feature itself, but should work in practice.}

However, you should consider that this particular method has relatively abysmal support (only ~66% of all browsers implement it as of this writing).