There are two ways you can assign Managers so that they can put their Specialty bonus to use:

• open the Managers tab while you are in the menu for a specific area of the bank (Marketing Office, Vault Room, Service Area)
or
• open the Managers menu (the cards icon bottom right of the bank) and open the 2nd tab (second icon from the right in the menu at the bottom of the Managers screen) - this is the "Assignments" section

If you see an unlocked slot available in either of these screens, you can tap it and a list of available Managers will open.

You can check which Managers are the most suitable for a given task (their Specialty) because they will be marked with a green icon. Tap "Assign" to put them to work.

As you increase your Reputation level, more Manager slots will become available.