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When somebody is going through a difficult life situation, people will commonly say, "We're pulling for you."

Where did this term come from? It sounds rather strange!

A Google Books search finds two examples of the phrase from the 1890s and two more from the very early 1900s. From "Johnnie You've Lost," reprinted from the San Francisco Examiner as part of "The Sketch Book—Character in Outline," in Current Literature (February 1890):

Both [bare-knuckles fighters] were winded and blood was flowing in streams. As soon as the men were in their corners the human stake [a waitress to be claimed, along with a gold ring, as a prize by the winner] lavished encouragement on Buckner, and as he was returning to the scratch cheered him with the remark:

"Don't give up, Johnnie; I'm pulling for you."

From Edgar Fawcett, "Old Uncle Vanderveer," in Outing (May 1895):

“So you won't help me, Uncle?” he muttered, sadly though not sullenly. “You won't give me a pull, so to speak?"

“One would suppose I'd never done any kind of pulling for you till now, Larry," came the answer, blent with a mildly sarcastic smile.

From Arthur McIlroy, "Pan-American Babies," in National Magazine (August 1901):

"I like the idea of naming a child for a good man. Many's the youngster that has Bill Daly for a handle to the name he inherited. Their mothers would ask me for the privilege! Think of that, my boys, when the privilege was all the other way. I'd say to myself, 'Bill, old man, here's another little one pulling for you—pulling you in the right direction.'"

And from the "Abbeville, N.C." report in Typographical Journal (August 1, 1902):

The convention committee is grateful for the support it is receiving , and is especially pleased with the letter received from Charlotte. We are glad to know that our neighbors are with us and pulling for us. We have a worthy ally in Charlotte, and with her help will make "The Old North State" heard from.

Two other interesting matches come from the years 1904–1904. From George Ogden, Tennessee Todd: A Novel of the Great River (1903):

"Andrews, as I said, is strong. He's got a strong combination pulling for him, too. The department don't doubt his ability to carry out the contract, but it does doubt the ability of his trains to make as good all-around time, up and down, and all considered, as the boats. ..."

And from "Beta-Epsilon: University of Wisconsin," in The Caduceus of Kappa Sigma (December 1904):

One of the greatest magnets which tends to draw prominence to a Fraternity and give it strength, too, is the athletic fraction of college life. We certainly have the magnet pulling for us "with all fours." Bro. Vanderboom, half-back on the foot-ball team, in the recent elections to the Athletic Board secured the Presidency, and Bros. Archie Pearsons and Art. Miller received prominent positions.

A much earlier example uses the phrase "pulling for him" not in the sense of "pulling on his behalf" but "pulling in direct competition with another for possession of him." From Mary Vidal, The Cabramatta Store; A Tale of the Bush (1850):

"Just so, Mat. Not that I care, I am sure, which [girl] Parry takes ; but 'tis fairly spoiling the young man to have two girls pulling for him, as you may say. However, they say he hasn't been very steady, and there are others have more claim on him now than either of these girls: but, hark! don't I hear the rails down ? Aye, and here she is!"

Unlike in this last example, the sense of "pulling for [someone]" in the other examples is "supporting [someone's] efforts spiritually or materially or both." Less clear is what the the metaphorical pulling originally referred to. Candidates include participants in a tug-of-war (as Josh61 suggests), a horse or other dray animal hitched to a plow or wagon, a person rowing or plying an oar, and a magnet. Of these possibilities, the dray animal and the rower may have been the most familiar images to contemporaneous audiences, but I haven't found any source that convincingly resolves the question.

I think it may come from 'Tug of war' as The Phrase Finder suggest:

Pulling for you:

• I am guessing that it comes "rope pull," a group activity at picnics, etc.
  This looks like what we call "Tug o' War" in Britain. Is that the case?
  Yes, that's what we in the U.S. call it too. But I just couldn't think of the term.

pulling for someone or something: (from TFD)

• to support and cheer for someone, a group, or something. We're pulling for you. We know you can do it! All the students were pulling for the team.

• the idea is to help someone in of a difficult or demanding situation especially giving moral support and the analogy with the 'Tug of War' activity, where supporters ideally help pulling the rope, is very effective in this context.

Could its origins be in the theater? I'm just guessing, but I think it could be someone pulling the curtain rope in the wings of a theater--a performer couldn't do that for herself--she would need someone "to pull for her". Can anyone confirm this, with evidence?

Could it have to do with the definition of “pull” that is “[TRANSITIVE] if a performer or a performance pulls an audience, a large number of people come to watch them”. So it would be someone shouting outside the circus tent, to “vouch” for you, vote for you, verbally gather attention and votes for you. And their reputation is on the line for you. That would get you the same impact that word of mouth would get. A fast, large, trusting audience.
Source for definition here: https://www.google.com/amp/s/www.macmillandictionary.com/amp/dictionary/british/pull_1

Tug of war?? Pulling a vehicle, farm animal, etc., out of the ditch or the mud? Pulling a ballot box lever?

Or could it have to do with ringing the bell at a church in recognition of a funeral or wedding, where you are pulling the rope for the bell? (See the Wikipedia entry for "Church bell").

Hemingway quotes part of John Donne's Devotions upon Emergent Occasions (1623), Meditation XVII in the epigraph of For Whom the Bell Tolls:

No man is an island, entire of itself; every man is a piece of the continent, a part of the main. If a clod be washed away by the sea, Europe is the less, as well as if a promontory were, as well as if a manor of thy friend's or of thine own were: any man's death diminishes me, because I am involved in mankind, and therefore never send to know for whom the bell tolls; it tolls for thee.