As per the instructions, its mentioned that we need to use the profile and audience for customer visitor id but as we don't have the privilege to use the people core service/AAM therefore can anyone please guide us the alternative way to enable the trigger through Adobe Launch.
I'm not sure I understand where you are in the process. To get access to the Triggers UI, in Admin Console, under an Analytics product profile there's a settings, that you need to click on for the profile itself, it should have a services tab on the left. Then from there, you should see a Triggers Service, you need to enable that. Then make sure you are a user of this profile with the service enabled. From there you can try logging in again and click on Launch from the Solution selector in the top right and you should see the option for Triggers UI. (This is all assuming your organization has had Triggers provisioned might want to check with your CSM if you don't see the triggers service as an option).
From there you can create a trigger in the UI, it is based on the experience cloud Id service meaning that it sends the ECID value along with whatever else you designate to campaign. So, you need to make sure you have that implemented and that campaign has some reference of that value also.