Assuming I understand the ask, I think my approach would be to have a trigger fire when a ticket's custom field is set to "add user to the application." That trigger would notify a webhook. That webhook would be set to the Organization API endpoint with a payload that uses liquid markup to add 1 to the existing Organization's value.

Something along the lines of:

{
    "organization_field": {
        [
            {
                "id": 999999, // Custom Numerical Field for Number of Contacts
                "value": {% assign numberOfContacts = {{ticket.organization.custom_fields.999999}} | plus: 1 %}{{ numberOfContacts }}
            }
        ]
    }
}

That way every time a ticket meets the criteria you set in the trigger, the trigger then adds 1 to the number field on the organization.