How to transfer my vault from an account owned by my former partner to my own 1Password account?

1. While they may be the Family Organizer, it's your 1Password account, so you can do whatever you want with your account and your vaults including your private vault. Any items in shared vaults which you want to retain will need to be copied to one of your vaults before your account disconnects from the Families plan.
2. Notwithstanding the above, I strongly recommend that you export all your data to secure storage. You should not need to import anything, but this insures you against that case.
3. Everything should be exported in a 1PUX and importing into 1Password should be more reliable that other clients which can also import 1PUX data.
4. Ensure that your data exists only in your vaults before disconnecting from the Families plan. 

I also suggest waiting for a response from 1Password staff before relying completely on my advice, though you may need to be patient. Take care of that 1PUX export and copying items from shared vaults while waiting.

1Password should add a "from" section to Migrate your existing 1Password data to a family account.