Partridge's A Dictionary of Slang and Unconventional English (2002) says:
into (a person) for (a sum of money), be. To owe a person so-much, to have let him down for a stated amount: Can, coll.: late C. 19-20. John Bearnes, Gateway, 1932, He's into me for ninety dollars, and I can't get a cent out of him.'
Where Can coll. is Canadian colloquial. The 2007 edition more simply says:
into preposition 1 in debt to, US, 1893
"... into me for ..."
Diving into Google Books, here's a possible 1902 from the American Ainslee's Magazine, Volume 10, Issues 1-6:
"Old man," confided Bill, after explaining the situation, "I need just a dollar and ten cents. Let me have it, like a good fellow." "But, Bill," replied Reece, hesitatingly, "you're into me for fifty dollars already."
And a definite 1903 from Arthur Morris Binstead's Pitcher in Paradise: Some Random Reminiscences, Sporting and Otherwise, published in London:
Two unplaced's an' one second, an' damme, she was into me for thirty-eight quid ! Stupid ? Aye, laad, even the bloomin' clerk rounded on me !
Here's a possible 1890 in Puck magazine which may be using a with a pun on the phrase:
"You 've got into me for all I 'm worth," remarked the Stocking to the Jumping-Jack. "All the same, I'm in a hole," replied the Jumping-Jack. And when Santa Claus heard them talking in that way, he broke the Jumping-Jack and took the ...
Walked and dribbled
Here's an interesting one from a possibly 1903 Pearson's Magazine:
He has dribbled into me for a thousand if he's had a cent, and now he must pay back by taking a chance.
Dribbling can also be found in the possibly 1917 Norsk-Engelsk Ordbog:
summer little sums ; cont driblets, petty sums. Cold has dribbled into me for a thousand ;
And finally, these two have walked into me for a [sum], which could be part of the same phrase.