I found some more details on her disappearance, as well as what sounds like a very similar-sounding case from two years later.
A fuller account of 'Elexis' disappearing (her name seems to be spelt differently in every story, which makes things trickier)
here:
Albany Child Lost
New-York Tribune, August 27, 1919
The Police Department yesterday received word from Chief of Police James L. Hyatt, of Albany, asking that search be made, especially in all gypsy trains passing through the city, for Elexis Stockburger, a twelve-year-old girl, who disappeared from her home, 12 Chestnut Street, Albany, on August 14. A reward of $500 is offered for her recovery and $1,000 for information leading to the arrest of her abductors.
A gypsy caravan passed through Albany on the day of her disappearance. She is small for her age, has black eyes, dark hair and a slight scar above the left eye. She wore a green gingham dress, black patent leather belt, white stockings, brown shoes and a blue coat.
Another from four days later
here ('Albany Girl Still Lost After 4 Days'):
No trace has been discovered beyond the assertions that two men saw a child answering her description riding through the city in an automobile with gypsies.
But it also mentions that the gypsy theory has been 'scouted' (from the context, I'm thinking this means 'scrapped' or 'discarded'?):
Suspicions that gypsies have kidnapped the child are scouted on the grounds itinerants have plenty of their own offspring and would not select for ransom a white child whose parents are not beyond moderate financial circumstances.
Then in 1923,
this one, titled 'Stockburger Girl Missing For Four Years To-Morrow: Mother, whose hair has turned gray, firmly convinced that daughter is dead'. This one describes her disappearance in more detail, and there's even a picture of her although the scan quality is too poor to really work out her features.
The child, as had been her custom, left the Chestnut street home at about 11:45 o'clock on the morning of Aug. 9 1919, to meet her sister, Esther, at Washington avenue and Swan street. She never met her sister, nor did anyone see her after she left her mother at the front stoop. Her mysterious disappearance is just as puzzling to-day as it was a half hour after she left her home, and a general search immediately after failed to reveal the slightest trace.
But she's also mentioned in
this story from March 1, 1921, about another disappearance of a girl in Albany: 10-year-old Harriet Menten, who was last seen heading for a school on the same street where Elixes was last seen.
The article about Harriet also mentions a man "loitering" outside Harriet's school a few weeks before she disappeared, who had been "seen to try to talk with some of the girls and boys" and "had attempted to engage two little girls in conversation". The article continues
here, under the title 'Mystery Marks':
Late this afternoon a man between 55 and 60, who said he lived on Hawk street, and who refused to tell the police his name was taken to police headquarters to be questioned. Three pupils at school 2, according to the police, identified him as the man who had tried to talk with some of the pupils. The man told the police that he looked after several furnaces on Chestnut street. He said that he liked children, and for that reason had talked to them.
After considerable questioning the man said his name was John Hoffman, and that he lived in the basement at 3 South Hawk street. He admitted, according to the police, that he knew the Menten girl, but denied that he saw her yesterday. Later Captain Lasch told Hoffman that he could return to his home.
Harriet was last seen setting off for school, and her mother says in that article that her usual route to school was "down Jay street, over Swan and down Chestnut". Alexis was last seen setting off to meet her sister, heading from Chestnut Street to the corner of Washington Avenue and Swan Street.
I can't find anything else about Harriet Menten, so maybe she turned up safe after all? It's strange she wouldn't have been mentioned alongside Elexis in that later article otherwise. But if she didn't, I can't believe that the cases aren't related.