Arwen didn’t go to the Undying Lands because she renounced this ability and chose not to go. However, she would use this decision as an argument for something else at the end of Lord of the Rings.
It is commonly noted that as part of the half elven family line, she had the choice to accept either the gift of mortality for the Men or Númenóreans or the gift of immortality for the Elves, and she chose the former.
However, she also gave her “ticket” to Frodo. At the time she saw how much he was suffering. Arwen would use this renunciation of her right to go west as an argument for Frodo to be able to do so.
“But the Queen Arwen said: ‘A gift I will give you. For I am the daughter of Elrond. I shall not go with him now when he departs to the Havens; for mine is the choice of Lúthien, and as she so have I chosen, both the sweet and the bitter. But in my stead you shall go, Ring-bearer, when the time comes, and if you then desire it. If your hurts grieve you still and the memory of your burden is heavy, then you may pass into the West, until all your wounds and weariness are healed.” (ROTK, “Many Partings”)
But it wasn’t explicit how exactly she would arrange this, as she would not be in direct communication with the Valar at the time.
One must have blessing of the Valar in order to enter the Undying Lands. Also, for any except the Elves – sailing west is not permitted and any exceptions required authority.
However, it can be presumed that Arwen either would arrange this before choosing a mortal life, and while still having this choice, she knew she wanted to renounce her ability to enter the West.
Also, have she not given her spot to Frodo, Arwen’s fate would still be decided by her choice either to be an immortal elf or to be a Númenórean and live a mortal life. This would also be what happens to Elrond’s brother Elros, who would die after ruling – but at a large lifespan compared to mortal men – and would not go to the West first.
Tolkien must’ve written Arwen (after deciding to be mortal) the same way he would with Aragorn and Elros, and the other earlier Númenóreans.
“…the early Númenóreans, though being fallen, by a special grace generally approach nearest to the unfallen Quendi, particularly in their right relations to God, to incarnate existence and self-mastery, to the natural world, and in their interests and arts.”
(NoME, App. A. “Metaphysical and Theological Themes”)
Gandalf would sail west as well – due to his being an emissary of the Valar and being in special accord with Cirdan the Shipwright.
It is also noted, however, that Arwen was the one who first thought of Frodo’s sailing west – and communicated through Gandalf to arrange it, as eventually he would accept her plea.
And after the death of her husband Aragorn, Arwen makes this choice clear.
“Nay, dear lord,” she said, “that choice is long over. There is now no ship that would bear me hence, and I must indeed abide the Doom of Men, whether I will or I nill: the loss and the silence.” (LOTR, App A)
