I will take my best stab at this, as I have been trying to figure it out myself as well. Street Fighter is not the first to do this. I have seen similar feats in Chinese wuxia television shows. In the old wuxia shows, the martial artists tend to shoot something more akin to lighting bolts in appearance. One is "Demi-Gods and Semi-Devils" (天龍八部). The television show I watched was from 1982 (predating Street Fighter a few years) while the books they are based on are from 1963. In Chinese wuxia lore, these magic blasts are used by martial artists, but primarily immortal daoists and other heavenly beings. These magic blasts are a manifestation of Dao and can be used to attack or heal. (This is in slight opposition to [or similiar to, depending on your personal belief as to what Dao is] Street Fighter and Dragonball Lore, where it is Ki, life energy.) [Trivia: If you play a Chinese video game RPG, rather than MP for your magic points, it is often labeled Dao and all magic use is Dao.]

Where this all originated from is the big question. The oldest source would be 西游记 "Journey to West" (1592) In this novel, the Buddhist and Daoist gods could shoot lighting bolts, fire and entrapping beams of light. It is a bit different than a hadoken, perhaps, but it is very close. If there is anything older, I don't know of it.

I absolutely agree with /u/DaLaohu that the influence for the hadoken was taken from Chinese cultural touchstones. Definitely Journey to the West was one. I wanted to give you more information and detail, however, on how those ideas came to be transmitted to Japan and give you a broader idea of the cultural milieu going on at the time.

So, for the hadoken, we thankfully have some accounts from people involved in creating Street Fighter. Takashi Nishiyama, the designer of the original Street Fighter as well as a host of other popular fighting games series for SNK, had this to say on the hadoken in an interview with 1UP:

Yeah, those [the fireball and dragon punch] were mine. 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 Shoryuken and Tatsumaki Senpuukyaku were original ideas -- I took martial arts moves and exaggerated them so they looked like special attacks.

So, from this we know he was directly inspired by Space Battleship Yamato. If you are unfamiliar, Space Battleship Yamato was a hugely influential anime that was immensely popular about a large battleship and the struggles of its crew as they venture through space. In it, the titular spacecraft had a large laser cannon that was fired to dramatic effect (Hadouho).

But, how do we get from laser cannons on a spaceship to a man firing a fireball? As the other post noted, this was not an original invention and perhaps Nishiyama was not even aware of the zeitgeist he was channeling at the time.

To better understand it, you have to take a look at anime and manga from the 70s-80s, specifically what is often described as “shonen manga.” Popularized in Weekly Shonen Jump, these works were targeting primarily young men and produced a number of hugely popular series in the period right around when Street Fighter came about. To name a few: Dragon Ball, Ranma ½, Captain Tsubasa, Mazinger Z, Slam Dunk, and Speed Racer. Many of these were imported and quite successful internationally (though some of them heavily altered).

What these works share is a “heightening” of action. For instance, this clip of Slam Dunk (https://www.youtube.com/watch?v=8EtUn0zVmWM) demonstrates it aptly: a simple outlet pass into a failed slam dunk takes over two minutes and includes many cut-aways to the protagonists friends, love interest, and team mates. The physical motions are exaggerated with lots of “slow motion” action.

What does all of this have to do with a fireball? Quite simply, some of these Shonen anime took Chinese myths and stories and combined it with the “heightening” and repackaged them for young males of the era. Dragon Ball did this and based itself heavily on “Journey to the West,” essentially becoming a retelling of the story. Goku, Dragon Ball's main character, ends up learning the Kamehameha Wave, a “ki blast” very similar to the Hadouken, during the “Tournament Arc” which aired in anime form not long before Street Fighter was made.

Now, anime and video games certainly formed a feedback loop, as can be seen from this panel of Ranma ½ that came out not long after Street Fighter. Two characters end up learing “ki blasts”, see this rather humorous panel of them ineffectually blasting one another while the girl they are squabbling over watches bemused (http://bp3.blogger.com/_svPQ_U3jni8/R5rd1h2FSLI/AAAAAAAAAQA/7Kvu3Nxrokc/s1600-h/RM20-045.GIF). Rumiko Takahashi seems to be poking fun at this trend of men blasting each other and even pokes fun at the name with her version the Shishi Hokodan.