How to store ansible_become_pass in a vault and how to use it?

You need to create some vaulted variable files and then either include them in your playbooks or on the command line.

If you change your inventory file to use a variable for the become pass this variable can be vaulted:

[my-servers]
san-francisco ansible_host=san-francisco ansible_ssh_user=user ansible_become_pass="{{ sanfrancisco_become_pass }}"
san-diego     ansible_host=san-diego     ansible_ssh_user=user ansible_become_pass="{{ sandiego_become_pass }}"

Then use ansible-vault create vaulted_vars.yml to create a vaulted file with the following contents:

sanfrancisco_become_pass: <my_sudo_password_for_user_on_san-francisco>
sandiego_become_pass    : <my_sudo_password_for_user_on_san-diego>

Then either include the vaulted file as extra vars like this:

ansible-playbook -i ~/.ansible/inventory/hosts playbook.yml --ask-vault-pass -e@~/.ansible/inventory/vault_vars

Or include the vars file in your playbook with an include_vars task:

- name        : include vaulted variables
  include_vars: ~/.ansible/inventory/vault_vars