I have two very nice tribbles that give marks (Dyson and Nikara). If I issue them to crew members will I still get the marks when they are used? From playing it does not seem so.
Unlikely... and since they can be used once every hour, your best bet is to have them on your character as they are likely to come off cooldown when your not in an area that you have whatever NPC they are equipped to available to you. The IDIC Tribble on the other hand, yes you do get that buff if it is equipped to an NPC.
1 mark every hour is pretty weak unless you play nonstop. A typical mission that lasts 5-10 min gives nearly 50 -- more than a standard work week's worth of marks from tribbles. I would put them aside.
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