If you are not familiar with modules, check out Ansible - Getting Started with Modules.

AVOID TROUBLE

Creating and removing logical volumes can only be done by root or by users or groups that have been granted sudo access, thus you'll want to either run this playbook as root, use the remote_user: root directive or use become.

The lvol module can be used to extend a Logical Volume, like this.

---
- hosts: all
  tasks:
    - name: extend Logical Volume lv001 in Volume Group vg001 to 8 GB
      lvol:
        vg: vg001
        lv: lv001
        size: 8g
...

However, you probably want to first determine if the Volume Group has enough available extents, which means you'll want to

1. Determine the current size of the Logical Volume
2. Determine the available extents in the Volume Group

Something like this.

---
- hosts: all
  tasks:
  - name: use lvdisplay to get the size of the /dev/mapper/rootvg-lv_var Logical Volume
    shell: lvdisplay /dev/mapper/rootvg-lv_var | grep "LV Size" | awk '{print $3}'
    register: lvsize

  - debug:
      var: lvsize

  # ---------------------------------------------------------------
  # This should return a string, almost always 'resizeable'
  # ---------------------------------------------------------------
  - name: use vgdisplay to get the available free extents in the rootvg volume group
    become: yes
    become_user: root
    shell: vgdisplay rootvg | grep 'VG Status' | awk '{print $3}'
    register: rootvg_status

  - debug:
      var: rootvg_status.stdout

  # ---------------------------------------------------------------
  # This should return something like 128 or 256 or 512, et cetera
  # ---------------------------------------------------------------
  - name: use vgdisplay to get the available free extents in the rootvg volume group
    shell: vgdisplay rootvg | grep 'Free  PE / Size' | awk '{print $5}'
    register: rootvg_free_extents

  - debug:
      var: rootvg_free_extents

  # ---------------------------------------------------------------------------
  # If rootvg_free_extents is 128, then rootvg_free_extents_float will be 1.0
  # If rootvg_free_extents is 256, then rootvg_free_extents_float will be 2.0
  # et cetera
  # ---------------------------------------------------------------------------
  - name: set_fact rootvg_available_extents_float
    set_fact:
      rootvg_available_extents_float: "{{ rootvg_free_extents.stdout|int / 128 }}"

  - debug:
      var: rootvg_available_extents_float

  # --------------------------------------------------------------------------------------
  # If lvsize.stdout is 7.00, then needed_extents will be 1.00 because 8.00 - 7.00 = 1.00
  # If lvsize.stdout is 6.00, then needed_extents will be 2.00 because 8.00 - 6.00 = 2.00
  # If lvsize.stdout is 5.00, then needed_extents will be 3.00 because 8.00 - 5.00 = 3.00
  # et cetera
  # --------------------------------------------------------------------------------------
  - name: set_fact needed_extents
    set_fact:
      needed_extents: "{{ 8.00 - lvsize.stdout|int }}"

  - name: fail if the rootvg Volume Group does not have enough available extents and is not resizeable
    fail:
      msg: The rootvg Volume Group has {{ rootvg_available_extents_float|int }} available extents thus the /dev/mapper/rootvg-lv_var Logical Volume cannot be extended to 8.00
    when: rootvg_available_extents_float|int < needed_extents|int and rootvg_status.stdout != 'resizeable'

  - name: extend Logical Volume lv001 in Volume Group rootvg to 8 GB
    lvol:
      vg: rootvg
      lv: lv001
      size: 8g
...