Update: I've asked producer Stuart Cohen and director John Carpenter about this. I'll amend my answer if and when they reply.
Despite the fact that Copper had the keys at one point, there is a much more important incident prior to that. When Split-Face is moved to the storage room, Garry gives Windows the keys to the storage room so he and Bennings can move the gear out of the room. They proceed to do so, and while Windows is away for a moment, Split-Face assimilates Bennings.
If you watch the scene in which Windows discovers Bennings in the middle of being assimilated by the Thing, and turn the volume up, you'll hear Windows drop the keys to the blood storage cooler. He doesn't pick them up again.
When the survivors find the blood cooler open, and the bags of blood destroyed, they accuse one another of being the culprit, but there is simply no way to determine who was really responsible.
I think it's safe to assume that the guilty party was a Thing. As such, here are our candidates:
Bennings-Thing: Although it was still in the process of assuming the form of Bennings, it was clearly the best placed to find the keys. However, I don't think it had either the motivation or the time to get the keys, go to the blood cooler, destroy the blood, and then flee the building. Because the men only realize that the Thing replicates its victims after they find Bennings-Thing, the blood test idea had yet to be raised when Bennings-Thing was killed. Thus, aside from the problem of the lack of time, Bennings-Thing also had no reason to destroy the blood supply.
Norris-Thing: We don't know when Norris was assimilated, but this is plausible. He knew that a blood test was in the works, so he had every reason to tamper with the blood supply.
Palmer-Thing: He was assimilated very early on, knew about the blood test, and had the time and motivation to destroy the blood supply.
Blair-Thing: Again, we don't know when he was assimilated, but he was a biologist, so he knew the potential use for the blood. However, he was locked up in the tool shed by the time the blood was destroyed, so he probably didn't do it.
We can rule out most or all of the following:
Windows: Not a Thing (he is later attacked by Palmer-Thing).
Nauls: Not a Thing (he is later attacked by Blair-Thing).
MacReady: Not a Thing.
Copper: Not a Thing (he is later attacked by Norris-Thing); came up with the idea of a blood test; discovered that the blood had been destroyed.
Childs: Almost certainly not a Thing.
Clark: Not a Thing (he is later killed by MacReady).
Garry: Not a Thing (he is later attacked by Blair-Thing).
In the category of "could have been a Thing, but probably wasn't", we have only one person:
The Best Explanation:
John Carpenter has never commented on this issue, as far as I know, and if he was asked to do so, he would almost certainly say "I don't know", as he does in response to pretty much every question about the movie. Therefore, as in most cases, the best source of information about this subject is the fansite Outpost 311.
The following comes from that site's FAQ section:
Q. Who got to the blood and how?
A: This is a question that has no definite answers, but there is a trail of actual events and logic that brings about some strong speculations.
The keys, which play a crucial role to the sabotaged blood scene, disappear from the action when Windows discovers the Bennings-Thing in mid-assimilation. He had gone to get the keys from Garry, came back, dropped them and ran out of the room. We don't see the keys show up again until right before the scene with the sabotaged blood. Someone obviously found them, used them, and somehow got them inconspicuously back into the action. (A good guess would be by placing them in or near Garry's quarters.) Watch Windows' face as they argue about who had access to the blood. His guilt from dropping the keys and not speaking up about it is a nice touch.
As for who did the deed, when Copper discovers the sabotaged blood it is still pouring out of the fridge, suggesting a very recent incident. (Contrast this to the script and novel where the blood is already fully dried.) Both Palmer and Norris are not present after locking up Blair when the men come up with the idea for a blood serum test. Perfect time to hit the blood supply. The other remaining men, Windows, Nauls, and Clark were most likely at their respective areas. (Clark distracted with his dead dogs and Windows probably recovering from his recent beating.) These two, Palmer and Norris, most likely Things now working together, were up to no good.
1 How reliable is the information on the Outpost 31 site? So reliable that, when the prequel was being made, and it was discovered that the studio hadn't preserved any blueprints for the Norwegian base set, the director of the prequel turned to Outpost 31, and was quickly provided with an exact blueprint of the set by the fans, based on careful examination of the relevant scenes from the 1982 film.