Is it safe to run Logic ICs at their maximum output current?
In the case of the 4094, the datasheet
  (https://assets.nexperia.com/documents/data-sheet/HEF4094B.pdf) lists
  the maximum input/output current at 10mA.


Let's be clear about the current you can supply from the 4094: -



Yes, the IO current is 10 mA but you can only drive 5 lines at 10 mA before you hit power supply current problems and, in any case you should not be driving the IO lines at 10 mA because that is their absolute maximum rating AND, without current limiting resistors this could easily be exceeded by your LED load. You should not assume that the IO lines are current limited at 10 mA in other words.

Regarding the LEDs, 30 mA is the absolute max rating and you should not approach it. I would say 20 mA is a fairly reasonable and safe limit as per this graph in their data sheet: -