How can I delete an old Apple ID that I no longer have the password to?

See if you can recover the password. Enter the Apple (iCloud) ID, and click Forgot Password. See if you can get a Rescue or Notification email option if necessary. Link to contact Apple Support at the end of the 2nd article.

Apple ID - Manage

Apple ID – Forgot Password  *      

If you have 2 Factor Authentication enabled, if you can't recover the password using the steps above, you will need to use Account Recovery. The Recovery Process is an automated process and the length of time it takes can vary a lot. Apple ID Support can only answer your questions. They can't intervene in or speed up the process. The article below has some suggestions to shorten the time the process takes.

Apple ID - Recover your Apple ID with two-factor authentication account recovery.   *

Click ▶︎ Apple ID has been compromised. 

You are welcome. Glad to help.