Remove support under interior stairs

I'd go ahead and pull it up and see what you've got. It sounds like it's in pretty poor shape so you're going to have to get rid of it either way.

Depending on how the carpet was installed you'll probably find tack strips that look something like like this:

You might also find staples -- I don't know if this is common but our installer used a staple gun in some troublesome spots. Most if not all of the staples will probably be pulled out when you rip out the carpet.

Before removing anything, take a look at the hardwood stair treads and make a preliminary decision about whether you think they're in good enough condition to leave exposed. If they're really rough and/or you want to recarpet, leave the tack strips in place and hammer in the staples.

If you do want to leave the stairs uncarpeted, you can just remove the tack strips, being careful not to damage the floor underneath. For minimal damage I'd recommend this type of prybar available at box stores for ~$14:

Use a hammer to tap the pointy claw end (like you would hammer a chisel) under the head of each nail holding the tack strip down, then pry up the nail enough that you can get your hammer's claw under it to pull it out. No tools ever need touch your stair tread.

( Note: I see your comment that this is no longer an issue, but since the question is still opened and unanswered, I'll put something down here to get it off of our list and possibly help someone else with a similar issue. )

First things first, Tester is right, don't pay the contractor until you get the permit, and force them to fix their mistakes at their own cost (regardless of whether you paid them or not).

Next, I'm going to suggest ignoring the premise of the question and you remove the stairs to fix this. Depending on how it's attached (I've only done interior stairs where ground movement wouldn't be an issue) you may be able to detach it from the deck, remove any bolts holding it to the supports, and take it out in one piece. Before removing the stairs, make sure to mark the location for the center of your cement pours using stakes about 4' away in two directions (one to the side, and one straight out to the yard, so that with a couple tape measures you can locate the exact spot).

For the hole, make it deep enough to get below the frost line in your location. Fill the bottom with gravel and pour your cement. I like the idea of metal brackets in the cement that keep the wood out of contact with the ground, rather than embedding the post in the cement itself, but only if you have a vertical load and no shear or twisting forces (e.g. I wouldn't use that for a fence). Make sure the bracket is above ground (my neighbor buried his and has all kinds of moisture issues). If you do place the wood support in the concrete, then at least make sure the concrete comes above ground level and slopes away from the wood on all sides.

Finally, reinstall your stairs. Hopefully you measured twice so that all you need to do is cut your supports to the right height and maybe drill some new holes for the bolts.