In order to survive, we're going to assume 3 necessities: food, shelter and reproduction. I'll deal with these in reverse, since it's convenient to leave the biggest issue - food - till last.
For reproduction, we know from the Silmarillion (chapter 3) that:
the Orcs had life and multiplied after the manner of the Children of Iluvatar
From that we can deduce that there are Orc-wives, Orc-babies and Orc-nurseries, and it's probably best to leave it at that.
For shelter, the subterranean settings you're asking about cover that sufficiently well enough, so we'll pass on to food. What can we find out about food?
In RotK (book 6 chapter 2) we find the following:
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 Nurnen; 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.
OK, so this is in reference to Mordor itself, not the Misty Mountains, but it does establish (1) that the Dark Lord's soldiers need food, (2) a means by which they get it in one location, and (3) a network of communication and commerce (of a sort).
Looking at the Silmarillion again, this time chapter 10, we find this:
the Orcs came down upon either side of Menegroth, and from camps in the east between Celon and Gelion, and west in the plains between Sirion and Narog, they plundered far and
An obvious question here is: "what are they plundering?" and even though Tolkien doesn't say so explicitly, we can imagine that food is included here.
It doesn't seem necessary to pull out more quotes; here we have 3 examples of ways that Orcs can get food to survive: slave-farms, tribute/commerce and plundering/raiding. It shouldn't be too much of a leap of imagination to think of other Orc strongholds - in Moria or Mount Gundabad, say - engaging in similar practices.