Okay, I really didn't intend to end up answering my own question, but this really piqued my curiosity so I dug into it.
The answer - weirdly - may actually be Spider-Man.
There does not seem to be any association between goblins and the colour green before the early 1900's. Searching shows that the early examples are fairy stories, which peaked in the 1920's. Then the association goes quiet until the mid 60's, when Spider-Man first encountered the Green Goblin.
Spider-Man and his green antagonist were created by Stan Lee and Steve Ditko, who were both born in the 1920's. Given this is when "green goblin" fairy tales were at a peak, and, apparently, via a school study aid at that, this is where they may have first formed the association. That said, as user @DaveSherohman points out, the authors were also fond of alliterative names — see Peter Parker, Jonah Jameson and so on — so it may just be a co-incidence.
Tolkien originally labelled his antagonists as goblins, and continued to use a loose association with the word when he switched to "orcs". The popularity of his work took off in the mid-60's — exactly the time when Spider-Man was re-enforcing the association between goblins and green.
Comic art has obviously always been an important influence on gaming culture generally. And when gaming exploded onto the scene in the mid-70's, there seem to have been two independent sources of green orcs and goblins.
The first source of green orcs seems to have been Tolkien artist Tim Kirk. He contribured this picture, The Road to Minas Tirith, to a 1975 Tolkien calendar:
And has been consistent in his portrayal of orcs as green:
The second is D&D. Orcs and Goblins were in the original 1974 rules but were not given a physical description. In the 1977 Monster Manual, however, they were described thus:
Description: Orcs appear particularly disgusting because their
— brown or brownish green with a bluish sheen — highlights their pinkish snouts and ears. Their bristly hair is dark brown or black,
Green was popularised by a colour partly via this 1980 Grenadier miniatures set:
So by the time Games Workshop came on the scene with Warhammer in 1983, the greenness of orcs seems already to have been well-established in gaming.