Мой вариант без применения методов массива (только разрешенный push)
var getDocumentsNumbers = function (documentsName, year) {
var counter = 0;
var yearToString = String(year);
for (let i = 0; i < documentsName.length; i += 1) {
var documentName = documentsName[i];
var documentYear = [];
for (let j = documentName.length - 2; j < documentName.length; j += 1) {
documentYear.push(documentName[j]);
}
if (documentYear[0] === yearToString[2] && documentYear[1] === yearToString[3]) {
counter++;
}
}
return counter;
}