A "request" is a formal document that you fill out and give to a company, a school, a government agency, etc. to ask for something.
In the example above, a "reimbursement request" is a document you fill out to ask your company to pay you back.
To "submit" a request means to give or send it to the people who need to receive it. You can also use "submit" for things like:
- homework assignments
- contest entries (like your artwork or a video you made)