Some organizations have a formal process for requesting (asking for) things. For example, if you work at a university, the university may have a system for asking for more staff.
In situations like these, you "put in a request" to ask for something. "Putting in a request" means that you submit your request in the correct way.
Other examples of "putting in a request" include:
- asking your boss for permission to take a long vacation
- asking your company to move to a different branch
- asking your local homeowner's association for permission to renovate your house
There are also other things that you can "put in", including:
put in an application for something
put in a reimbursement request (to get paid back for money that you spent for your job)
put in a resignation (which means to formally quit your job)