$all_items = [
  $a1 = [ 'name' => 'item1', 'description' => 'item 1', 'item_key' => 123 ],
  $a2 = [ 'name' => 'item2', 'description' => 'item 2', 'item_key' => 456 ],
  $a3 = [ 'name' => 'item3', 'description' => 'item 3', 'item_key' => 789 ]
];

$invoice_items = [
  $b1 = [ 'item_key' => 123, 'desc' => '1item' ],
  $b2 = [ 'item_key' => 456, 'desc' => '2item' ],
  $b3 = [ 'item_key' => 123, 'desc' => '3item' ],
  $b4 = [ 'item_key' => 345, 'desc' => '4item' ],
  $b5 = [ 'item_key' => 123, 'desc' => '5item' ],
  $b6 = [ 'item_key' => 345, 'desc' => '6item' ],
  $b7 = [ 'item_key' => 123, 'desc' => '7item' ]
];

$invoices = [];
foreach ($invoice_items as $invoice_item) {
  $invoices[$invoice_item['item_key']][] = $invoice_item;
}

$result = [];
foreach ($all_items as $all_item) {
  $result[] = array_merge([ $all_item ], $invoices[$all_item['item_key']] ?? []);
}

The structure of the resulting array is the same as you described in your question above.

But there is a contradiction between your result set and your comment "...merge the invoice_item array into a new array". Your result array has the invoice items appended, hence my code above.

Now, this solution does create a new key - here called "descs" - an array in the invoice items under this key:

$invoices = [];
foreach ($invoice_items as $invoice_item) {
  $invoices[$invoice_item['item_key']][] = $invoice_item;
}

foreach ($all_items as &$all_item) {
  $all_item['descs'] = $invoices[$all_item['item_key']] ?? [];
}
unset($all_item);

Since the item_keys in these "descs" array entries are redundant, you can remove them and just create a string array:

$invoices = [];
foreach ($invoice_items as $invoice_item) {
  $invoices[$invoice_item['item_key']][] = $invoice_item['desc'];
}

foreach ($all_items as &$all_item) {
  $all_item['descs'] = $invoices[$all_item['item_key']] ?? [];
}
unset($all_item);

Output:

Array
(
    [0] => Array
        (
            [name] => item1
            [description] => item 1
            [item_key] => 123
            [descs] => Array
                (
                    [0] => 1item
                    [1] => 3item
                    [2] => 5item
                    [3] => 7item
                )

        )

    [1] => Array
        (
            [name] => item2
            [description] => item 2
            [item_key] => 456
            [descs] => Array
                (
                    [0] => 2item
                )

        )

    [2] => Array
        (
            [name] => item3
            [description] => item 3
            [item_key] => 789
            [descs] => Array
                (
                )

        )

)

Please note that your input arrays all have assignment statements in their construction ($a1 = ..., $a2 = ..., etc.). This doesn't make much sense unless you need these variables later. If so, encoding is more readable and better:

$a1 = [ 'name' => 'item1', 'description' => 'item 1', 'item_key' => 123 ],
$a2 = [ 'name' => 'item2', 'description' => 'item 2', 'item_key' => 456 ],
$a3 = [ 'name' => 'item3', 'description' => 'item 3', 'item_key' => 789 ]

$all_items = [ $a1, $a2, $a3 ];