Edited from a previous answer:

Let's talk about definitions: the genre of music we're discussing gets called 'rap' sometimes, but a lot of people prefer the word 'hip-hop', because 'hip-hop' denotes a certain culture and set of practices that go together beyond just simply the rapped vocals that are a key characteristic of the genre (but not the only key characteristic). To belabour the point, you get plenty of 'hip-hop' music that is instrumental, without featuring rap - for example, DJ Shadow's 1996 album Endtroducing record is entirely based on samples, that were used in a very hip-hop kind of way. It sounds basically like 'rap music' except that there's no-one rapping over the top. Ultimately the history of hip-hop as a genre starts in the 1970s when DJs in Brooklyn, New York like Kool Herc, were looping 'breakbeats' from soul/funk records (a fair few records had instrumental drum-only sections where the drummer had a brief place to shine, that get called 'breakbeats'), using the kind of sound manipulation techniques used in the sound systems of Jamaica (Kool Herc had relatively recently arrived in New York as an immigrant from Jamaica). Amongst these DJs, a tradition emerged after a year or two of MCs rapping over the breakbeats assembled by the DJs, and so a hip-hop DJ like Grandmaster Flash began to bill himself as 'Grandmaster Flash & The 3 MCs'. 'Rappers Delight' by the Sugar Hill Gang in 1979 was the first record to effectively harness this kind of mix of elements into a pop single.

And that's hip-hop! But you can have rap in music which doesn't claim to be hip-hop. The alternative rock band R.E.M., for example, have a song called 'Radio Song' which features a rap from the rapper KRS-One. But you'd be hard pressed to call R.E.M. hip-hop, or even 'rap music'. The reason I point this out is because rap music is not just about the rapping! So where there might be an influence of 'flyting', it's specifically an influence on the practice of rapping, rather than hip-hop in general.

But back to rhyming, kind of rhythmic, spoken word performances: this existed before the hip-hop genre. There's a tradition popular from the medieval era in Western Europe - mentioned everywhere from Beowulf to Chaucer to Shakespeare - of ritualised poetic insult trading, called flyting. There are also, worldwide, a bunch of similar traditions - including, I suspect repente, which I don't particularly know anything about. Between them, these traditions are definitely something of an ancestor of the tradition in rap of freestyle competitions where rappers try to diss each other for the amusement of bystanders (most famously depicted in the Eminem movie 8 Mile).

Anyway, to get from medieval flyting to rap, we need to look at the tradition that more directly influenced rap: 'the dozens'. A recent book by Elijah Wald, Talkin' Bout Your Mama, discusses 'the dozens' at length: it's more or less an extended, elaborate 'yo mama' joke-telling competition. You can get a sense of what 'the dozens' sounds like on this 1959 Bo Diddley track, 'Say Man', which features Diddley and his guitarist trying to insult each other, dozens-style, for fun and profit. If Rudy Ray Moore did develop his particular rhyming style by learning it from homeless men in the 1960s and 1970s, the dozens is likely what those homeless men were doing.

The dozens is a very obvious precursor for rap - it's African-American people in the 20th century rhyming, often in a somewhat musical way. And it's likely, says Wald, that Scottish versions of the 'flyting' tradition influenced the dozens, in terms of some of the turns of phrase and rhyming techniques, via Scottish immigrants to America who would have come in contact with African-Americans. However, Wald also points out that ritualised insult trading traditions can be found everywhere from Polynesian atolls to North Africa; 'flyting' in the Anglosphere context is not particularly unique. And Wald argues, much more strongly, that African insult-dueling practices, rather than medieval Northern European flyting, has had much more of an influence on the formation of rap.

It's also important to recognise that other forms of spoken word poetry have directly influenced the idea of rapping over music, beyond the 'dozens' tradition. There is a large influence of Jamaican sound system culture in the way that hip-hop has developed - especially in the way it sounds and the way it recontextualises pre-existing records. That culture also has a tradition of 'toasting', where DJs chant in a monotone over the music; as some early hip-hop DJs in New York in the 70s had Jamaican roots, there is an obvious influence of toasting on rap which is separate to the American tradition of the dozens.

Similarly, there is also an obvious influence of spoken word poetry, and in particular the quite literary traditions of jazz poetry and beat poetry, on rap. This is most obviously personified in the 1971 track 'The Revolution Will Not Be Televised' by Gil Scott-Heron, and his contemporaries the Last Poets. It doesn't quite have the rhythmicness of rap, but Gil Scott-Heron comes from a more literary tradition involving Langston Hughes's 'jazz poetry' (e.g., this 1958 performance of 'The Weary Blues', where Hughes recites poetry over jazz) or perhaps Jack Kerouac's experiments with beat poetry over music.

If you look at 'Rapper's Delight' by the Sugarhill Gang, it has the Big Bank Hank verse dissing Superman which is obviously influenced by the dozens:

I said he's a fairy I do suppose

Flying through the air in pantyhose

He may be very sexy or even cute

But he looks like a sucker in a blue and red suit

I said you need a man who's got finesse

And his whole name across his chest

He may be able to fly all through the night

But can he rock a party 'til the early light

He can't satisfy you with his little worm

But I can bust you out with my super sperm

However, in contrast, the social message in Grandmaster Flash and the Furious Five's 'The Message' comes from a more obviously jazz poetry/Gil Scott-Heron/Last Poets-esque kind of place. For instance, Duke Bootee, who wrote the lyrics of 'The Message' and rapped on much of it, has a website where he proudly quotes a journalist's comparison of his rapping to Langston Hughes.

Rudy Ray Moore's tall, overtly sexual, tales of Dolemite, generally, fit into that dozens kind of vibe, but I also suspect that Snatch and the Poontangs' (absurdly profane) 1969 self-titled album is an influence on the records Rudy Ray Moore was putting out in the early 1970s - Johnny Otis's vocals on the Poontangs record are a little more melodic/sung, but also aren't that far away from rapping, and the profane tall-tale telling is very likely an influence. Also, there are obviously similarities between Snatch and the Poontangs' 1969 'Signifyin' Monkey (Part 1)', and Rudy Ray Moore's 1970 'Signifying Monkey'. The other thing I notice, listening to bits and pieces of what Rudy Ray Moore was doing in the early 1970s, was that his rhyming was not as highly rhythmic as modern rap is - it's spoken words over music, but it doesn't have the focus on the beat that, say, 'The Message' has - it's more like a less highbrow descendant of what Langston Hughes was doing, in some ways.

So I suspect that if the movie of his life makes a claim for him to be the Godfather of Rap, it's a bit of a tall tale - there's lots of different influences on the music. But if it's basically claiming one influence of a few on people who became rappers, then sure - no doubt plenty of the teenagers in Brooklyn who created hip-hop had Dads who listened to Rudy Ray Moore and would have heard the music. But I note that in the obituary of Moore in the New York Times in 2008, the acts that are identified as having an influence of Moore are, e.g., Snoop Dogg, 2 Live Crew, Big Daddy Kane - artists that came to prominence in the late 1980s/early 1990s; Moore's profane/ribald tall tales were clearly an influence on the milieu that Snoop Dogg portrays in his raps, full of oversexed pimps and hustlers. But this kind of lyric-writing is definitely focused on the late 1980s and onwards - earlier rap styles in hip-hop were generally less ribald/more focused on partying.