When you use your company's employment terminal, you'll find new options for both employees and management.
This will collect a description of your reimbursement reason, the amount being requested as well as a brief 'budget code' for your part of the company (ask your boss). example budget codes: VS-RND, NLM-TV.
View your requests pending review, those that have been approved and those that have been resolved (paid or denied).
If you have the management rights on the employment terminal, you'll be able to approve and deny reimbursement requests via this option in the MANAGE EMPLOYEES sub-menu.
Approved reimbursements should pay out when you're able to collect your weekly pay. Note I say should as I have yet to test the change to payout, so please report bugs if you encounter them. :)