Prefix the string with the parts that determine the sort, i.e. yymmdd and 2 letters from the "ORDER" string, as it turns out that when you pick the 7th and 8th letter of those words (EA, ET, TE), they will be sorted correctly. Then after having sorted the items, remove that prefix again.

Here is how that works out:

let array = [
  'ORDERHEADER_010122.arc',
  'ORDERITEM_010122.arc',
  'ORDERDETAIL_010122.arc',
  'ORDERDETAIL_010222.arc',
  'ORDERDETAIL_010322.arc',
  'ORDERHEADER_010222.arc',
  'ORDERHEADER_010322.arc',
  'ORDERHEADER_010422.arc',
  'ORDERITEM_010222.arc',
  'ORDERDETAIL_010422.arc'
];

let sorted = array.map(item => 
    item.replace(/ORDER.(..).*?_(..)(..)(..).*/g, "$4$3$2$1") + item
).sort().map(s => s.slice(8));

console.log(sorted);

Extending it

If you have more prefix-words of which you want to control the order, then create an array of those in their expected order. The solution will then turn that array into a lookup map (giving a 4-character sequence number for a given word). The call to replace then needs a callback argument, that will make the lookup and prefixes that sequence. Here is the code for that:

let array = [
  'ORDERHEADER_010122.arc',
  'ORDERITEM_010122.arc',
  'ORDERDETAIL_010122.arc',
  'ORDERDETAIL_010222.arc',
  'ORDERDETAIL_010322.arc',
  'ORDERHEADER_010222.arc',
  'ORDERHEADER_010322.arc',
  'ORDERHEADER_010422.arc',
  'ORDERITEM_010222.arc',
  'ORDERDETAIL_010422.arc'
];

let priorities = [
    'ORDERHEADER',
    'ORDERDETAIL',
    'ORDERITEM',
];

// Map the priority array to an object for faster look-up
let priMap = Object.fromEntries(priorities.map((word, i) => 
    [word, ("000" + i).slice(-4)]
));

let sorted = array.map(item => 
    item.replace(/(.*?)_(..)(..)(..).*/g, (all, word, dd, mm, yy) =>
        yy + mm + dd + (priMap[word] ?? "----") + all
    )
).sort().map(s => s.slice(10));

console.log(sorted);

You have to define a custom compare function for your sorting method call. And, that method should first compare dates, and then (if a date is the same) order prefixes according to your requirements

Here is my example

const order = new Map() // one can use plain Array + Array#indexOf later on
  .set('ORDERHEADER', 0)
  .set('ORDERDETAIL', 1)
  .set('ORDERITEM', 2)

const compareFn = (a, b) => {
  const [a1, a2] = a.split('_')
  const [b1, b2] = b.split('_')

  const r2 = a2.localeCompare(b2) // TODO: re-write to date comparison a2 vs b2

  if (r2 !== 0) return r2

  return order.get(a1) - order.get(b1) // or Array#indexOf as mentioned above
}

// usage
array.sort(compareFn)