get_all_files.gd

func get_all_files(path: String, file_ext := "", files := []):
	var dir = Directory.new()

	if dir.open(path) == OK:
		dir.list_dir_begin(true, true)

		var file_name = dir.get_next()

		while file_name != "":
			if dir.current_is_dir():
				files = get_all_files(dir.get_current_dir().plus_file(file_name), file_ext, files)
			else:
				if file_ext and file_name.get_extension() != file_ext:
					file_name = dir.get_next()
					continue

				files.append(file_name)

			file_name = dir.get_next()
	else:
		print("An error occurred when trying to access %s." % path)

	return files

If you want the full path of the file instead of just the name, change:

Line 17

files.append(file_name)

for

files.append(dir.get_current_dir().plus_file(file_name))

Upgraded version for godot 4.x

func get_all_files(path: String, file_ext := "", files := []):
    var dir = DirAccess.open(path)

    if DirAccess.get_open_error() == OK:
        dir.list_dir_begin()

        var file_name = dir.get_next()

        while file_name != "":
            if dir.current_is_dir():
                files = get_all_files(dir.get_current_dir() +"/"+ file_name, file_ext, files)
            else:
                if file_ext and file_name.get_extension() != file_ext:
                    file_name = dir.get_next()
                    continue
                
                files.append(dir.get_current_dir() +"/"+ file_name)

            file_name = dir.get_next()
    else:
        print("An error occurred when trying to access %s." % path)

    return files