Who is supposed to prepare inspection requests for completed or ongoing activities in a construction site?

No there is confusion in all these answers.

Suppose that a single party is involved in the project. The job will be done by the site engineer/ site supervisor. Then the site engineer will request the quality controler to come and inspect his job. For this the site engineer will prepare the inspection request. It is very basic understanding that quality controler will check the job based on the site engineer's request. It is not that quality controler will raise the RFI and check his own request. 

In a project where multi parties are involved the contractor site engineer will raise the IR or WIR or 'RFI' and forward it to his QC for inspection. When his QC found it is ok, 

Then this contractor QC will take this IR to consultant/ client for inspection. So the Inspection request is always/ has to be prepared by site engineers not Quality controler.

A quality controler is always an invitee not an invitor. When multi parties are involved this RFI has a  'via' in quality controler. 

No there is confusion in all these answers.

Suppose that a single party is involved in the project. The job will be done by the site engineer/ site supervisor. Then the site engineer will request the quality controler to come and inspect his job. For this the site engineer will prepare the inspection request. It is very basic understanding that quality controler will check the job based on the site engineer's request. It is not that quality controler will raise the RFI and check his own request. 

In a project where multi parties are involved the contractor site engineer will raise the IR or WIR or 'RFI' and forward it to his QC for inspection. When his QC found it is ok, 

Then this contractor QC will take this IR to consultant/ client for inspection. So the Inspection request is always/ has to be prepared by site engineers not Quality controler.

A quality controler is always an invitee not an invitor. When multi parties are involved this RFI has a  'via' in quality controler. 

Construction supervisor has to prepare inspection requests and submit to inspector for inspection

quality control engineer supposed to prepare inspection requests for completed or ongoing activities in a construction site

It varies between Projects and sites. If you consider a site, then it is the responsibility of the Project Incharge at the site should request for inspections with relevant authorities. If seperate quality assurance engineer is present means ; Then He shall request for inspections with the authorization of the project Head at that specific site.