tl;dr: it is based on a weapon in the anime Space Battleship Yamato, with the in-universe source being chi, a concept of energy in Chinese martial arts and traditional medicine.
The designer of the first game, Takashi Nishiyama, talks about the specific move’s inspiration:
I was inspired by anime and manga from the time, so for example the Hadouken was inspired by a Japanese anime called Space Battleship Yamato. In that, the battleship has a laser missile called Hadouho — it collects energy and then blasts it into space, destroying the enemy. That that's where I got the idea for the Hadouken.
The name apparently is directly derived from the weapon in the show; 波動拳 ("wave motion fist") is the Hadouken and 波動エンジン ("wave motion engine" with "engine" in English) is the name of the weapon. A Wikia page on the weapon.
A picture of the Hadouho:
Wikipedia says this about the move:
The move is achieved by the character thrusting their palms forward, sending a surge of spirit energy (or ki1) towards the opponent ("chi1 blast").
On “chi”1, a supposed energy or life force in Chinese traditional medicine, Wikipedia says:
Qi1 is the central underlying principle in Chinese traditional medicine and in Chinese martial arts.
The practice of controlling one’s chi1 is called Neijing.
The in-universe explanation for this is using chi1, a concept in real-world Chinese martial arts.
1Note that all of these are the same thing; "ki" in Japanese is the same thing as "chi" or "qi" in Chinese