Morgoth had a lot of creatures inside Thangorodrim while it was besieged and blockaded by Elves for many years. In later years his army broke out, so could more easily have found food, but for all those years besieged, where did the food come from? Thangorodrim itself was rubble and slag and rock, no good for growing food. To the north was an encircling ring of blocking mountains (with apparently some secret pathways out but they were a long way around), and to the south were armies of Elves. Morgoth had a very large army of orcs to feed, plus huge dragons and other beasts. How did they get fed?
The army didn't live in Thangorodrim. Thangorodrim was the gatehouse guarding the tunnel under the Iron Mountains leading to the underground fortress and kingdom of Angband.
Apparently the underground fortress of Angband had uncounted levels and as many square miles or kilos of surface area in those levels as a good sized nation. And it would have had a dense population since Morgoth would use highly advanced technologies to produce food, etc. for his population of orcs, etc.
Modern techniques like hydroponics and aeroponics can grow several times more food per square unit of land than modern dirt farming which can grow several times more food per square unit of land than old fashioned medieval farming could. And they don't need natural sunlight. Artificial light will do. Gimli's song claims that the Dwarves of Moria had artificial light during the First Age.
Or Angband may have used more advanced methods of chemical synthesis of food. Thus every square mile or kilo of barren ice covered surface of Angband covered an undergrund region that could support as many persons as tens, or hundreds, or thousands of square miles or kilos on the surface.
I don't think it's ever explained in canon.
However, Angband/Thangorodrim was not completely encircled, even during the Siege of Angband. The Noldor couldn't or didn't block it off to the North.
The Orcs could have done terrace farming in the Iron Mountains or maybe even Thangorodrim itself, possibly, depending on how cold they were.
We don't have a clear picture of the world north of Angband - if the First Age equivalent of the Arctic Ocean was close enough, they could have fished or whaled.
Also, there seem to have been some raiding parties (eg into Hithlum) even during the Siege, so maybe they captured food or hunted in warmer lands too.
EDIT: Also, the Siege lines probably weren't drawn up to the gates of Angband itself. Morgoth controlled Dor Daedeloth, a land outside the gates of Angband at Thangorodrim, which was only rarely entered by the Elves : e.g. at the first rising of the Sun
"At the uprising of the great light the servants of Morgoth fled into Angband, and Fingolfin passed unopposed through the fastness of Dor Daedeloth while his foes hid beneath the earth. Then the elves smote upon the gates of Angband, and the challenge of their trumpets shook the towers of Thangorodrim"
"There upon the confines of Dor Daedeloth, the land of Morgoth, Feanor was surrounded, with few friends about him."
"But the Noldor being again united set a watch upon the borders of Dor Daedeloth"
(all from Silmarillion Chapter 13 - Of the Return of the Noldor)
The later battles were largely fought on Ard-galen/Anfauglith, south of Dor Daedeloth. It seems pretty clear that Morgoth's forces controlled more than just the subterranean fortress/prisons of Angband itself.
The food was brought in.
The simplest explanation lies in this passage, a few moments after Sam's rescuing Frodo and both escaping the tower of Cirith Ungol. Emphasis mine:
Frodo and Sam gazed out in mingled loathing and wonder on this hateful land. Between them and the smoking mountain, and about it north and south, all seemed ruinous and dead, a desert burned and choked. They wondered how the Lord of this realm maintained and fed his slaves and his armies.
Neither he nor Frodo knew anything of the great slave-worked fields away south in this wide realm, beyond the fumes of the Mountain by the dark sad waters of Lake Núrnen; nor of the great roads that ran away east and south to tributary lands, from which the soldiers of the Tower brought long waggon-trains of goods and booty and fresh slaves. Here in the northward regions were the mines and forges, and the musterings of long-planned war.
The lord of the Rings, chapter The land of Shadow, second book of The Return of the King
Your assumption that Mordor is completely encircled by either ennemies or natural impasses is not completely right.
Ithilien, the wooden lands going between the great river Anduin and Mordor's mountain, were getting more and more under Sauron's domination. From the East, the mountains did not exist and although the map doesn't show it, many roads existed as well.