I went through and found the quotes from the episodes/movies listed for Kirk:
Where No Man Has Gone Before:
KIRK: If you have not received a signal from me within twelve hours, you'll proceed at maximum warp to the nearest Earth base with my recommendation that this entire planet be subjected to a lethal concentration of neutron radiation. No protest on this, Mark. That's an order. -
KIRK: That's an order, Mister Spock.
The Deadly Years:
KIRK: That's an order, Lieutenant. Code two.
The Return of The Archons:
KIRK: That's an order. Watch him. Captain out.
KIRK: That's an order, Scotty. -
This Side of Paradise:
KIRK: Spock, I don't know what you think you're doing, but this is an order. Report back to me at the settlement in ten minutes. We're evacuating all colonists to Starbase Twenty Seven.
Who Mourns For Adonais?:
KIRK: And one more thing. I want no more unauthorised action against Apollo or whatever he is. That's an order!- ( and later)
KIRK: Fire those phasers! That's an order, Mister Spock! -
The Immunity Syndrome:
KIRK: No, no, no. That's an order.
The Devil in the Dark :
KIRK: You're a healer. There's a patient. That's an order. Mister Spock. Tell it we're trying to help. The mechanism.
The Cloud Minders:
KIRK: Officially, yes. But this mission is strictly unofficial. Nobody's to have any part in it, any responsibility for it, but myself. That's an order, Spock. You have the conn. Stand by till I contact you.
The Lights of Zetar:
KIRK: Get to Sickbay, Lieutenant. That's an order.
TAS The Infinite Vulcan:
KIRK: Out of my way, mister. That's an order.
KIRK: (to comm) Whatever happens, Mister Sulu, if you think you have the tactical advantage, you fire on that ship, even if we're still onboard. That's an order.
ST: Into Darkness:
Sign for the torpedoes, that's an order.
And enough with the metaphors, all right? That's an order.
My friend and I couldn't find where Kirk said it in The Trouble with Tribbles, so if someone could point it out for me, it would be appreciated.
Also Archer said it 16 times not 15.
- ARCHER: Give me the phase pistol. That's an order.
- ARCHER: At this point, it won't make much of a difference. You two get to a shuttlepod. They're only after humans. If this doesn't work out you'll still have a chance. That's an order.
- ARCHER: Clap louder. That's an order.
'Strange new worlds':
- ARCHER: Put your weapon down, Trip. That's an order.
ARCHER: We're wasting time. Help me figure out a way to get you off of here. That's an order. (and later)
ARCHER: I'm not going to argue with you, Malcolm. It was twenty. That's an order.
- ARCHER: That's an order. No offence, but my ears are less likely to draw fire than yours.
- ARCHER: Don't try to tell me who started it! I don't know what the problem is between you two, and to tell you the truth I don't care, but I want it settled right now. Settle it! That's an order!
- ARCHER: Get some help if you need it, but make this a top priority. That's an order.
10 .ARCHER: You drink it. That's an order.
- ARCHER: Name them. That's an order.
'Shadows of P'Jem':
12 .ARCHER: I didn't ask if you were hungry. You need to keep up your strength, so eat. That's an order.
- ARCHER: Not another word, Commander. That's an order.
- ARCHER: Five bars of gold? Open these things, Sub-Commander. That's an order.
- ARCHER: Take a nap, Trip. That's an order.
- ARCHER [OC]: That's an order.
So unless someone can find Kirk's That's an order in The Trouble with Tribbles, Kirk and Archer are even with 16 each.