Is there a way to save voicemails to iCloud

If you do not have live Voice Mail, unfortunately, iPhones don't directly store voicemails in iCloud or on the device itself. Voicemails are typically stored on your carrier's servers.

Forward Voicemail via Email or Message:

• Open the Phone app and tap Voicemail.
• Tap the desired voicemail.
• Select the Share icon.
• Choose how you want to forward the voicemail:
  • Mail: Attach the voicemail to a new email.
  • Messages: Send the voicemail as an audio message to a contact.

Store Voicemail in iCloud

You can store and sync your Live Voicemail and Carrier Voicemail in iCloud.

1. Go to Settings  > [your name] > iCloud.
2. Tap Show All under Apps Using iCloud, then turn on FaceTime.

Source: Set up your voicemail on iPhone – Apple Support (MY)