No. There is no need to buy a new external drive to complete a Time Machine backup. If there is enough space left on your external hard drive for a backup, there’s nothing stopping you from using it. However, make sure that the disk is formatted correctly. Otherwise, you’ll be prompted to erase the data stored there and reformat it.
Note that if you’re using an internal drive, it should be partitioned into two volumes before the backup. And alternatively, if you wish, you can create a network backup.