Okay, I think you’ve got a little bit of a vocabulary mix-up here, because enlisting and volunteering (at least in the US military) are the same thing. The fanon/canon is that Bucky was drafted—essentially, his name/number was called and he had the choice to go, run away to Canada, or be imprisoned.
The reason this has really taken hold is because the service number he repeats while in Zola’s lab in the first movie is “32557(038).” The “32” at the beginning indicates that he is from New York, and that he was drafted (a “12” at the beginning would mean he enlisted).
While it’s possible an intern had a case of the Wikipedias and just picked a number off the Wiki page, this seems unlikely for a couple reasons. Number one, it’s actually a line in the film, so it was the screenwriters (Markus and McFeely) who wrote that in. Number two, that same Wikipedia page has the explanation of the numbers right next to the image, and the explanation is much, much easier to read than the image. Number three, the service number on Steve’s documents from the first film start with “12,” which means whoever put those together was aware of what the numbers indicated (even if the props department gets basically everything else wrong, ha!).
I’ve seen the clip where Sebastian Stan talks about his theory re: Bucky’s enlistment. He essentially says that he thinks Bucky enlisted out of a sense of duty—to the country, to his family, to his father’s memory—and then found that war was not what he thought it would be (that’s a broad paraphrase). I actually really like this headcanon (and I love that Stan has his own theories on his character), but I still think he’s wrong.