As seen in the posts of those who quoted Appendix A and B, it is only stated that "it is said" Gimli went West, so there is not certainty to it to begin with, but it can be taken as accurate. Also, In Letter 154 Tolkien writes that Gimli went to Aman "as friend of Legolas and 'servant' of Galadriel."
I don't really think that Legolas had the ability to get Gimli into the west, but the rekindled friendship between dwarf and elf was an important feat that may have helped influence the Lords of the West. Galadriel was much more important than Legolas, but she had also undergone a falling out with the Noldor (In Letter 246 Tolkien writes that Galadriel (whose lock of hair Gimli recieved as a gift) wouldn't have influence because of her staying in Middle-Earth, but this in reference to Frodo's departure. She would have regained the Valar's favor after her return to Aman, although this might not mean enough influence).
Also, being Gandalf's friend would have helped him earn the favor of the Valar, as he is of fundamentally the same race, although less powerful as a Maia. Furthermore, after his return from death he is given pretty much a role as the messenger of Eru, who sent him back and enlarged his role in Middle-Earth.
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"He was sent by a mere prudent plan of the angelic Valar or governors; but Authority had taken up this plan and enlarged it, at the moment of its failure."
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