Yes, there are documents and third party projects indicating that it has a free public API, but I haven't been able to get it to work. I presume that a paid API would have better availability and the possibility of support.

I just tried waybackpy and I'm getting errors with it too when I try to reproduce their basic demo operation:

  >>> from waybackpy import WaybackMachineSaveAPI
  >>> url = "https://nuclearweaponarchive.org"
  >>> user_agent = "Mozilla/5.0 (Windows NT 5.1; rv:40.0) Gecko/20100101 Firefox/40.0"
  >>> save_api = WaybackMachineSaveAPI(url, user_agent)
  >>> save_api.save()
  Traceback (most recent call last):
    File "<python-input-4>", line 1, in <module>
      save_api.save()
      ~~~~~~~~~~~~~^^
    File "/Users/xxx/nuclearweapons-archive/venv/lib/python3.13/site-packages/waybackpy/save_api.py", line 210, in save
      self.get_save_request_headers()
      ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~^^
    File "/Users/xxx/nuclearweapons-archive/venv/lib/python3.13/site-packages/waybackpy/save_api.py", line 99, in get_save_request_headers
      raise TooManyRequestsError(
      ...<4 lines>...
      )
  waybackpy.exceptions.TooManyRequestsError: Can not save 'https://nuclearweaponarchive.org'. Save request refused by the server. Save Page Now limits saving 15 URLs per minutes. Try waiting for 5 minutes and then try again.