How difficult would it be to come into a match trying to get a stalemate? Is it also possible to repeat this? Could one raise their elo with the ability to just stalemate every game?
First of all what is a stalemate? This is a specific kind of draw (you can have a lot of them: perpetual check, insufficient material, mutual agreement, threefold repetition, 50 moves rule and a stalemate).
But in comparison to other draws, stalemate is most of the time a blunder made by your opponent. This is because it enables a player with inferior position to draw the game. The blunder of such high quality that it costed him a draw in completely winning position.
So to answer your questions:
How difficult would it be to come into a match trying to get a stalemate?
Extremely difficult an quite silly. Because to get into stalemate you would need to reach in the beginning much worse position and only after that to hope that your opponent will do a big mistake. Yes, it is possible (some of GM's had it), but it is better to play for a draw in another way (where you do not put yourself in bad position in the first place)
Is it also possible to repeat this?
Sure, if your opponent do not want to learn and want to do the same queen stalemate every time.
Could one raise their elo with the ability to just stalemate every game?
Even theoretically it is impossible (even if you will assume that you can stalemate every game, which is ridiculous) because you will play with people above your rating and below your rating so for the draw with people below your rating you will get minus score.
If the goal of the game were stalemate instead, the game would still be more or less the same as chess.
In chess, in order to win, one tries to get some kind of superiority (material, space, position) and win from there by forcing the opponent to yield more and more until no resistance is possible.
In most chess variants, a key principle is to be able to force your opponent around by establishing some kind of superiority. (Exactly the same!) After all, once you have a winning position it is easy to do anything thereafter.
Indirect proof 1: Consider the variants antichess and losing chess (captures must be played if available, win condition is to lose all your own units; in antichess additionally kings are not royal.) It is well-known that getting a material advantage is almost always the correct way to play for a win. The resulting extra mobility eventually lets you force your opponent to take all your pieces in a single final burst, or to arrive at a winning material-up endgame.
Indirect proof 2: I cite the club correspondence game Paris - Marseille, 1878. Paris took white and started without a queen. Black's goal was to force white to deliver checkmate; white tried to avoid mating black. The game looks remarkably normal for a long way, until black's advantage is sufficient to start forcing white around.
I claim, if from an advantageous position one can force the opponent to deliver mate against their will, what more difficult would forcing stalemate be?
[Date "????.??.??"] [Result "*"] [FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNB1KBNR w KQkq - 0 1"] [White "Paris"] [Black "Marseilles"] [Event "correspondence 1878; white gives queen odds, black plays for selfmate"] 1.d4 d5 2.Nc3 c6 3.Nf3 g6 4.e4 e6 5.e5 Bb4 6.Bd2 Bxc3 7.Bxc3 b5 8.h4 h5 9.O-O-O a6 10.Ng5 f5 11.g3 Nh6 12.Bd3 Nf7 13.Bxf5 gxf5 14.Nxf7 Kxf7 15.Bd2 Nd7 16.Rhe1 c5 17.dxc5 Nxc5 18.Bg5 Qg8 19.Re3 Bb7 20.Rc3 Rc8 21.Be3 Nd7 22.Bd4 Rxc3 23.bxc3 a5 24.Kd2 a4 25.Rb1 Ba6 26.Rg1 Qg4 27.Rb1 Rc8 28.Rb4 Rc4 29.Rxc4 dxc4 30.a3 f4 31.Kc1 fxg3 32.fxg3 Qxg3 33.Kb2 Qxh4 34.Kc1 Qe1+ 35.Kb2 Qd1 36.Ba7 Nxe5 37.Bc5 h4 38.Bd4 Nc6 39.Be3 e5 40.Bf2 h3 41.Bg3 e4 42.Bf4 Ke6 43.Bg3 e3 44.Bf4 e2 45.Bg3 Kd7 46.Bh2 e1=Q 47.Bf4 Qee2 48.Bg3 Qdxc2+ 49.Ka1 Qf1+ 50.Be1 Qd2 51.Kb1 h2 52.Ka1 h1=Q 53.Kb1 Qf8 54.Ka1 Qxa3+ 55.Kb1 Qad6 56.Ka1 Qf6 57.Kb1 Kc7 58.Ka1 b4 59.Kb1 b3 60.Ka1 Kb6 61.Kb1 Ka5 62.Ka1 Ne7 63.Kb1 Nc8 64.Ka1 Bb5 65.Kb1 Qa6 66.Ka1 Nb6 67.Kb1 Qh7+ 68.Ka1 Qxc3+ 69.Bxc3# *
Getting an advantageous position (exactly the same as normal chess) would be the simplest way to force stalemate.