Polyfill for input type="month" with no dependencies.

input-month-polyfill.js

(function() {
	var monthInputs = document.querySelectorAll('input[type="month"]'),
		checkDateInput = document.createElement('input'),
		dateSupported = false,
		months = [],
		lang = document.documentElement.lang || navigator.language,
		DEFAULT_SPAN = 5;
		
	if (monthInputs[0].type === 'month') {
		// browser supports month input; no need for polyfill
		return true;
	}
	checkDateInput.setAttribute('type', 'date')
	dateSupported = 'date' === checkDateInput.type;
	
	function makeRangeArray(from, to) {
		var result = [], idx;
		if (!to) {
			to=from;
			from=0;
		}
		for (idx=from; idx < to; idx += 1) {
			result.push(idx);
		}
		return result;
	}
	
	if (!dateSupported) {
		// initialize months
		makeRangeArray(12).forEach( // generate size 12 array and loop over
			function (_, idx) { 
				var dt = new Date(1900, idx, 15);
				months[idx] = dt.toLocaleString(
					lang, 
					{'month': "long"}
				)
			}
		);
	}

	function createMonthDropdown(selected) {
		var monthDropdown = document.createElement('select');
		selected = selected || new Date().getUTCMonth();
		selected = selected.getUTCMonth ? selected.getUTCMonth() : selected;
		months.forEach(function (monthName, monthIdx) {
			var month = monthIdx + 1,
				opt = document.createElement('option');
			opt.value = month;
			opt.text = monthName;
			if (monthIdx === selected) {
				opt.setAttribute('selected', 'selected');
			}
			monthDropdown.appendChild(opt);
		});
		return monthDropdown;
	}

	function createYearDropdown(selected, dropdownYears) {
		var yearDropdown = document.createElement('select');
		selected = selected || new Date().getUTCFullYear();
		selected = selected.getUTCFullYear ? selected.getUTCFullYear() : selected;
		dropdownYears.forEach(function (year) {
			var opt = document.createElement('option');
			opt.value = year;
			opt.text = year;
			if (year === selected) {
				opt.setAttribute('selected', 'selected');
			}
			yearDropdown.appendChild(opt);
		});
		return yearDropdown;
	}
	
	function getYears(startYear, endYear, yearSpan) {
		var years;
		yearSpan = yearSpan || DEFAULT_SPAN // by default, get 10 years before & after
		if (!startYear && !endYear) {
			startYear = new Date().getUTCFullYear() - yearSpan;
			endYear = new Date().getUTCFullYear() + yearSpan;
		} else {
			startYear = startYear || endYear - (yearSpan * 2);
			endYear = endYear || startYear + (yearSpan * 2);
		}
		if (startYear > endYear) {
			throw new Error("Invalid inputs; end year must be less than start year");
		}
		years = makeRangeArray(startYear, endYear + 1);
		return years;
	}
	
	function getLastDayOfMonth(dt) {
		dt = dt instanceof Date ? dt : new Date(Date.parse(dt));
		dt.setUTCMonth(dt.getUTCMonth() + 1)
		dt = new Date(dt - 1);
		return dt.getUTCDate();
	}

	var YEAR_MONTH_PATT = /^\d\d\d\d-\d\d$/;
	monthInputs.forEach(function (monthInput) {
		var initial = monthInput.value,
			yearEl, monthEl, dateEl,
			yearSpan = parseInt(monthInput.dataset.yearSpan),
			yearDropdown = monthInput.dataset.yearDropdown,
			minMonth = monthInput.min,
			maxMonth = monthInput.max,
			startYear, endYear, years;
		if (minMonth) {
			startYear = parseInt(minMonth.split('-')[0], 10);
		}
		if (maxMonth) {
			endYear = parseInt(maxMonth.split('-')[0], 10);
		}
		if (typeof yearDropdown !== 'undefined') {
			yearDropdown = !(
				yearDropdown === false ||
				yearDropdown.toLowerCase() === 'false'
			)
		}
		if (initial && YEAR_MONTH_PATT.test(initial)) {
			initial += '-01';
		}
		initial = initial && new Date(Date.parse(initial));
		initial = (initial && initial.getTime && !isNaN(initial.getTime())) ? initial : new Date();
		monthInput.dataset.year = initial.getUTCFullYear();
		monthInput.dataset.month = initial.getUTCMonth() + 1;
		years = getYears(startYear, endYear, yearSpan);
		if (dateSupported) {
			dateEl = document.createElement('input');
			dateEl.setAttribute('type', 'date');
			dateEl.monthInputTarget = monthInput;
			dateEl.monthInputType = 'date';
			if (monthInput.min) {
				if (monthInput.min.split('-').length >= 2) {
					dateEl.setAttribute(
						'min', 
						monthInput.min + '-01'
					);
				} else {
					dateEl.setAttribute('min', monthInput.min + '-01-01');
				}
			}
			if (monthInput.max) {
				if (monthInput.max.split('-').length >= 2) {
					dateEl.setAttribute(
						'max', 
						monthInput.max + '-' + getLastDayOfMonth(monthInput.max)
					);
				} else {
					dateEl.setAttribute('max', monthInput.max + '-12-31');
				}
			}
			if (monthInput.value) {
				dateEl.value = (
					monthInput.value + '-01'
				);
			}
			monthInput.parentNode.insertBefore(dateEl, monthInput);
		} else {
			if (yearDropdown === false) {
				yearEl = document.createElement('input');
				yearEl.setAttribute('type', 'number');
				if (startYear) {
					yearEl.setAttribute('min', startYear);
				}
				if (endYear) {
					yearEl.setAttribute('max', endYear);
				}
				yearEl.setAttribute('step', 1);
				if (initial) {
					yearEl.value = initial.getUTCFullYear()
				}
			} else {
				yearEl = createYearDropdown(initial, years);
			}
			yearEl.monthInputTarget = monthInput;
			yearEl.monthInputType = 'year';
			monthEl = createMonthDropdown(initial);
			monthEl.monthInputTarget = monthInput;
			monthEl.monthInputType = 'month';
			monthInput.parentNode.insertBefore(monthEl, monthInput);
			monthInput.parentNode.insertBefore(
				document.createTextNode(' '),
				monthInput
			);
			monthInput.parentNode.insertBefore(yearEl, monthInput);
		}
		// monthInput.style.display = 'none';
	});
	
	function updateValue(input) {
		var date, newValue, monthDisplay;
		if (input.dataset.date) {
			date = new Date(Date.parse(input.dataset.date));
			newValue = (
				date.getUTCFullYear() + 
				'-' + 
				(date.getUTCMonth() + 1).toString().padStart(2, 0)
			);
		} else {
			newValue = input.dataset.year + '-' + input.dataset.month.toString().padStart(2,'0');
		}
		if (input.max && newValue > input.max) {
			monthDisplay = months[parseInt(input.max.split('-')[1])];
			if (monthDisplay) {
				alert(
					"You cannot choose a date later than " + 
					monthDisplay + ", " + 
					input.max.split('-')[0]
				)
			} else {
				alert(
					"You cannot choose a year greater than " + input.max.split('-')[0]
				)
			}
			return false;
		} else if (input.min && newValue < input.min) {
			alert("Value cannot be less than " + input.min);
			return false;
		}
		input.value = newValue;
		return true;
	}
	
	document.addEventListener('focus', function (event) {
		var focused = event.target;
		if (focused.monthInputTarget) {
			focused.dataset.storedValue = focused.value;
			if (typeof focused.selectedIndex !== 'undefined') {
				focused.dataset.storedIndex = focused.selectedIndex;
			}
		}
	}, true);

	document.addEventListener('change', function (event) {
		var changed = event.target;
		if (changed.monthInputTarget && changed.monthInputType) {
			changed.monthInputTarget.dataset[changed.monthInputType] = changed.value;
			if (!updateValue(changed.monthInputTarget)) {
				if (changed.dataset.storedIndex) {
					changed.selectedIndex = changed.dataset.storedIndex;
				} else if (changed.dataset.storedValue) {
					changed.value = changed.dataset.storedValue;
				}
				changed.monthInputTarget.dataset[changed.monthInputType] = changed.dataset.storedValue;
				changed.focus();
			} else {
				changed.dataset.storedValue = changed.value;
				if (typeof changed.selectedIndex !== 'undefined') {
					changed.dataset.storedIndex = changed.selectedIndex;
				}
			}
		}
		return true;
	}, true);
		
}())

Tried with Firefox 59, the result is: I'm getting an additional input date field, when I select a day from such input the input month is updated with matching month. Not exactly what I expected...