They actually did do so, just not on a particularly large scale! There is a famous photo, Reindeer Yasha at War that is set during the German assault on Murmansk (I say set, because the photo is a composite of multiple shots, which I talk about here). Anyways though, you'll note that the planes in the background are Hawker Hurricanes! The RAF sent an Air Wing, the 151, to the Eastern Front as a sort of goodwill gesture. Yevgeni Khaldei had become friendly with some of the pilots, and when he made his composite, he chose to use the Hurricanes instead of Soviet aircraft. 151st Air Wing remained there for a pretty short time, and returned to the UK by the end of the year, after giving their planes to the Soviets to use (showing the strengths of their aircraft, and training the Reds in their use was another part of their mission). This article seems to offer more information that I can though if you want a more detailed account of their time there, and the Wiki page isn't half bad either.

(Additionally a squadron of Free French pilots, the Normandie-Nieman, flew on the Eastern Front, but they were much more integrated with the Soviet Air Force, and flew Soviet aircraft like the Yak-3)/

The Americans tried to set up a small military mission to the Soviet Union in late 1941 - observers, trainers, and so forth - but this foundered due to Soviet suspicion and after lingering in occupied Iran for a few months the U.S. technical advisory team was dissolved.

Later in the war, the USAAF launched Operation Frantic. At the December 1943 Tehran Conference, Roosevelt agreed to supply U.S. bombers to support the summer Soviet offensive. In February 1944, U.S. support personnel began upgrading the Soviet airfield at Poltava. On June 2, 1944, a few flights of B-17s took off from Italy, hit targets in Hungary, and then landed at Poltava. Over t he next couple of weeks, the B-17s struck targets in Romania and Germany. A German recon plane shadowed the bombers back to Poltava - it had been a Luftwaffe base the previous year so the Germans knew well how and where to knock the base out of commission. On June 22, German bombers destroyed 47 of the 73 B-17s on the ground and damaged most of the rest.

With only nine operational bombers, Operation Frantic was drastically downscaled but attacks continued through September. At that point, the progress of the war in France and the difficulties of working with the Soviets led the USAAF to suspend operations. By the time winter weather lifted, the frontline was too far west for Poltava to be effective, and no Operation Frantic flights were launched in 1945.

EDIT: This only covers U.S. actions. Please see /u/Georgy_K_Zhukov's response for information on British and French activities.

Talking about the eastern front we may use a broader scope starting from 1939 and the Winter War between Finland and the Soviet Union. It is widely accepted that Soviets shelled their own village of Mainila, claimed that to be an aggressive act by the Finnish Army and declared war. During this time Germany, who had earlier trained Finnish Jaeger units that were instrumental in the white (non-communist) victory over the reds (communist) during Finnish Independence War was considered friendly towards Finns, but due to the secret Molotov-Ribbentrop pact and non-aggression between Germany and the Soviet Union no official aid could be given and the Reich did not allow troop and material transports through it's areas to Finland. At this time there were no "Allies" so to say, but the western nations were sympathetic to the Finnish cause and offered both material aid and thousands of volunteers to fight the war. There are some records online, but generally the aid was quite small scale compared to total material and human losses suffered and more in the line of motivational support. US opinions were predominantly anti-soviet in this cause and technically it could be said that the western allies stood against the soviets in the frozen forests of eastern Finland.

We Finns got our asses handed to us but managed to scare the bold russians. Peace terms were horrible for a small nation and reparations asked for a war we did not start were outrageous. General consensus was that the western nations, with the exception of Sweden, did little even though they had the power to do more and when Germany offered a chance of revenge and almost unlimited material support the Finnish political leaders did not hesitate much to take the deal. This led to de facto alliance (even though we prefer to call it brothers in arms or some bull...) between Finland and Germany, The Continuation War that started in 1941 as the northern arm of Operation Barbarossa, declaration of war from the Allies to Finland as they were now friendly with the soviet union, the fall of Germany and the Finnish front, peace terms with the Soviets and Lappland War in 1945 that saw Finnish troops fighting and driving out German units from Northern Finland.

The second part of my post is simplified in-a-nutshell version of Finnish involvement in WW2 and is debated to some extent specially in political motives -field. As for the original question of US and Commonwealth support on the eastern front, they supported the Finns fighting overwhelming odds and later when the German war machine started rolling in a way they did not like they supported Soviets fighting against Finns, who were now allied with Germany.

EDIT: Had to go through my sources as they are books:

(Colonel) Bauer; World War 2 (1966), volumes 1-6 - Finnish version produced by Colonel Mikola, Colonel Tervasmäki and Lieutenant Colonel Seppälä in 1973

Mann, Jörgensen; Hitler's Arctic War (2002)

Torbacke, Magnusson, Rosenfeldt, Weibull; The First And Second World Wars (1986, 2002) (Finnish Edition by Otava