I think LotR was only a minor influence to Gygax himself. . .Gygax seemed as much or more influenced by Robert E. Howard, Jack Vance, Fritz Lieber and Poul Anderson, and Michael Moorcock than by J.R.R. Tolkien.
Tolkien is a huge influence on the fantasy supplement for Gary Gygax and Jeff Perren’s wargame Chainmail (1971), which was based on a
Lord of the Rings wargame created by Leonard Patt. In fact I’d say his influence is greater than all of those other authors combined. The majority of troop types derive either wholly or partially from Tolkien. "Almost all of the
Chainmail monsters had indisputable and often unique originals in Tolkien" according to Jon Peterson's Playing at the World. Tolkien is the only fantasy writer to be mentioned by name three times in the text. (Howard, Anderson, and Moorcock appear once each.)
1974 OD&D is less Tolkien-esque than Chainmail (not because anything is removed but because OD&D adds many more elements from other sources) but I think Tolkien still has a greater influence than any other single fantasy author.
In the original version there were references to hobbits, balrogs, ents, “Nazgul of Tolkein [sic]”, “Barrow Wights (per Tolkein) [sic]” and “Eagles of Tolkein [sic]”.
These were all renamed after a 1977 cease-and-desist letter. In addition there are elves, dwarves, orcs, goblins, and werebears. Spells include hold portal, light, wizard lock, knock, lightning bolt and fire ball. Magic items include the ring of invisibility and elven cloak and boots.
Broader Tolkienesque themes include: the adventuring party which consists entirely of Tolkien races, and magical and non-magical members; groups of evil humanoid monsters (orcs, goblins, hobgoblins, kobolds, gnolls) inhabiting a large underground lair similar to the Mines of Moria in The Lord of the Rings or the Misty Mountains goblins in The Hobbit; the conflict between the aforementioned; the 'orc-ification' of folkloric creatures such as kobolds and hobgoblins turning them tribal and non-magical; the wargame aspects such as stronghold construction costs.
Considering the authors you mention, I think Anderson’s influence is the greatest because the alignment system is very important.
Anderson: Law vs Chaos (essentially good vs evil in OD&D), regenerating trolls
Vance: Spell memorisation
Howard: Evil high priests, ghouls, cloudkill
Moorcock: Intelligent swords
Leiber: Lightning bolt
The thief and barbarian classes, Moorcock-ian Law and Chaos, and more minor elements such as thieves' and assassins' guilds, and the spell prismatic spray, are later developments.
