In fact, Gill owned at least one Clissett ladderback himself. It can be seen (just, right at the bottom) in the photograph of Gill in his engraving shop in about 1940. Gill was a member of the Art Workers Guild for about five years in the early 20th century, resigning at about the same time as his sister married. He would have known Clissett's chairs from there.
How reliable is the story about the provenance of the set of Clissett chairs that Gill is supposed to have purchased? Well, the chair in Knell's photograph belonged to "a descendant of the original recipient". From Knell's photographic acknowledgements, we can see that this is probably "Mrs R. Stewart-Jones". A little research shows that this is actually the daughter of Gladys Gill by her second marriage. So the provenance is good, and the story reasonable, though it's not possible to corroborate the suggestion that Gimson was obtaining Clissett's chairs for others - essentially acting as an agent.
Except that we know that Gimson bought a good number of chairs from the Clissett workshop as late as 1914, so he clearly stayed in touch with the Clissetts and traded with them, despite having his own chair-making business.