Dart

canMove does a recursive search and returns all locations that need moving, or none if there's an obstacle anywhere downstream. For part2, that involves checking if there's half of a box in front of us, and if so ensuring that we also check the other half of that box. I don't bother tracking whether we're double-checking as it runs fast enough as is.

import 'dart:math';
import 'package:collection/collection.dart';
import 'package:more/more.dart';

var d4 = <Point<num>>[Point(1, 0), Point(-1, 0), Point(0, 1), Point(0, -1)];
var m4 = '><v^';

solve(List<String> lines, {wide = false}) {
  if (wide) {
    lines = lines
        .map((e) => e
            .replaceAll('#', '##')
            .replaceAll('.', '..')
            .replaceAll('O', '[]')
            .replaceAll('@', '@.'))
        .toList();
  }
  var room = {
    for (var r in lines.takeWhile((e) => e.isNotEmpty).indexed())
      for (var c in r.value.split('').indexed().where((c) => (c.value != '.')))
        Point<num>(c.index, r.index): c.value
  };
  var bot = room.entries.firstWhere((e) => e.value == '@').key;
  var moves = lines.skipTo('').join('').split('');
  for (var d in moves.map((m) => d4[m4.indexOf(m)])) {
    if (didMove(d, bot, room)) bot += d;
  }
  return room.entries
      .where((e) => e.value == '[' || e.value == 'O')
      .map((e) => e.key.x + 100 * e.key.y)
      .sum;
}

bool didMove(Point m, Point here, Map<Point, String> room) {
  var moves = canMove(m, here, room).toSet();
  if (moves.isNotEmpty) {
    var vals = moves.map((e) => room.remove(e)!).toList();
    for (var ms in moves.indexed()) {
      room[ms.value + m] = vals[ms.index];
    }
    return true;
  }
  return false;
}

List<Point> canMove(Point m, Point here, Map<Point, String> room) {
  if (room[here + m] == '#') return [];
  if (!room.containsKey(here + m)) return [here];
  var cm1 = canMove(m, here + m, room);
  if (m.x != 0) return (cm1.isEmpty) ? [] : cm1 + [here];

  List<Point> cm2 = [here];
  if (room[here + m] == '[') cm2 = canMove(m, here + m + Point(1, 0), room);
  if (room[here + m] == ']') cm2 = canMove(m, here + m - Point(1, 0), room);

  return cm1.isEmpty || cm2.isEmpty ? [] : cm1 + cm2 + [here];
}