The neatest way I can think of:
The param compare is just like the compare function used in Array.sort().
Sort child nodes.
/**
 * @param {!Node} parent
 * @param {function(!Node, !Node):number} compare
 */
function sortChildNodes(parent, compare) {
  const moveNode = (newParent, node) => {
    // If node is already under a parent, append() removes it from the
    // original parent before appending it to the new parent.
    newParent.append(node);
    return newParent;
  };
  parent.append(Array.from(parent.childNodes) // Shallow copies of nodes.
                    .sort(compare) // Sort the shallow copies.
                    .reduce(moveNode, document.createDocumentFragment()));
}
Sort child elements (a subset of child nodes).
/**
 * @param {!Element} parent
 * @param {function(!Element, !Element):number} compare
 */
function sortChildren(parent, compare) {
  const moveElement = (newParent, element) => {
    // If element is already under a parent, append() removes it from the
    // original parent before appending it to the new parent.
    newParent.append(element);
    return newParent;
  };
  parent.append(Array.from(parent.children) // Shallow copies of elements.
                    .sort(compare) // Sort the shallow copies.
                    .reduce(moveElement, document.createDocumentFragment()));
}