Did the Ring Bearers and Gimli die in Valinor?

Frodo is mortal, and going to Valinor doesn't change this. I don't remember this being explicitly stated in The Lord of the Rings. Book VI closes as Frodo sails away. Gandalf, Galadriel and the others do not make any prophecies regarding what will happen in Valinor. One of the appendices provides a timeline for “later events concerning the members of the Fellowship of the Ring”, but strictly limited to Middle-earth. The Ring is destroyed in 1422 by Shire reckoning, Sam sails West in 1482 when his wife dies, and Legolas and Gimli (“it is said”) are the last of the Fellowship to leave Middle-earth in 1541. That's all you'll find in The Lord of the Rings: that story is told from the point of Middle-earth and does not chronicle what happens in the Elven lands that Man cannot reach.

Mortality is a gift to Men (this is clearly established in the Simlarillion). This is a gift of Ilúvatar and not even the Valar can affect it. Although Hobbits are not mentioned, they are probably close enough cousins of Men to be mortal in the same way. The case of Dwarves may be less well-established but nothing indicates otherwise.

We can turn to Word of God in the form of The Letters of J.R.R. Tolkien. There it is clearly stated (letter 246):

‘Alas! there are some wounds that cannot be wholly cured’, said Gandalf — not in Middle-earth. Frodo was sent or allowed to pass over Sea to heal him — if that could be done, before he died. He would have eventually to ‘pass away’: no mortal could, or can, abide for ever on earth, or within Time. So he went both to a purgatory and to a reward, for a while: a period of reflection and peace and a gaining of a truer understanding of his position in littleness and in greatness, spent still in Time amid the natural beauty of ‘Arda Unmarred’, the Earth unspoiled by evil.

Letter 325:

As for Frodo or other mortals, they could only dwell in Aman for a
limited time — whether brief or long. The Valar had neither the power nor
the right to confer ‘immortality’ upon them. Their sojourn was a
‘purgatory’, but one of peace and healing and they would eventually pass
away (die at their own desire and of free will) to destinations of which
the Elves knew nothing.

There you have it: the Ring-bearers were mortal, they remained mortal, and they eventually died in the manner of mortals, albeit after a time of their own choice.

A final note, still in letter 325:

(…) But the legends are mainly of ‘Mannish’ origin blended with those of the Sindar (Gray-elves) and others who had never left Middle-earth.

As this last passage shows, we don't know precisely what happened to Frodo and the other Ring-bearers in Valinor because what we know is the legends of Middle-earth. We have no knowledge of what happened in Valinor after it was removed from this Earth, save through what little contact it still had with Middle-earth until the end of the Third Age.