From a previous post....
The main reason why they couldn't beam down blankets is because the transporters weren't working at all (ship-wide) for most of the episode. The Evil Kirk Double shot a big hole in the transporter systems' circuitry casing when Spock nerve pinched him:
Mr. Scott, sir, on the lower level of the
engineering deck. I found the new trouble
with the transporter. The casing has a
gap ripped in it. The main circuits have been
burned through. The abort control circuit
is gone altogether....The transporter unit
ionizer--Nothing much left of it, sir....We can't
repair it in less than a week.
So although they discovered that the transporters were duplicating people and things in the middle of Act One, by the middle of Act Two, it was a whole new ballgame with the transporters and they weren't working at all
Why didn't Kirk beam down a whole bunch of blankets in that short interval after they discovered the duplicating problem but before the transporters stopped working altogether? Well, because the "Good Kirk" was already losing his power of command and wasn't thinking clearly. In fact, even after the transporters circuitry was damaged by the phaser fire, Good Kirk never even thought to mention this new damage to Scotty. Here's some pretty interesting dialog from the actual Final Draft shooting script that didn't make it to the final episode:
Haven't you checked all the circuits yet, Scotty?
Yes, sir, we have. And we'd thought we found
the trouble. Now something else has gone wrong.
We don't know, sir. We're checking on it now.
And then after Scotty does finally track down the problem, here's what he has to say in the Final Draft shooting script:
Mr. Scott, sir. I'm on the lower level of the
engineering deck. I've just found the new
trouble with the Transporter. The Transporter
Ionizer Unit has been mangled! Looks as if a
phaser gun's been fired at it.
So, Kirk hadn't even mentioned the problem to Scotty, even though Kirk saw the whole thing happen and knew about the problem; clearly Kirk was already becoming forgetful, neglectful, and preoccupied. Poor Scotty had to hunt down the cause this new transporter problem even though Kirk saw it happen!
So, why were no additional blankets beamed down? Because during the very short window of opportunity they had to beam down any blankets, Kirk wasn't thinking straight (and then all the transporters stopped working altogether).
In looking at the terrain of Alfa 177, it looks pretty rocky with high crosswinds. I don't think there's a place to set a shuttlecraft down within hundreds of miles of Sulu's survey team. It might
have been possible, but as I mentioned,the Kirks weren't thinking very clearly that day. Even if using a shuttlecraft were possible, the poor, incapacitated Kirk "slowly losing the power of command" didn't think of it.