Sindar/Grey-elves were just like the rest of the Eldar in their immortality. Círdan the shipwright is Sindar and he sails to the undying lands on the last ship from middle-earth. Círdan would have been 10,000+ years old during the war of the ring. He was the oldest elf in Middle-earth in the Second age, being born sometime in the years of the trees.
When the elves awoke and were called to Aman the Sindar were those of the Teleri who elected to stay in middle-earth. In Beleriand Elu Thingol and his wife Melian the Maia founded the kingdom of Doriath. Their daughter Lúthien is the subject of the the "tale of Beren and Lúthien". The tale tells of the quest to retrieve a Silmaril so that Beren can marry Luthien. A task Thingol thinks will end in Berens death and thus keep them from marrying. Towards the end Beren is killed and Luthien through grief passes from middle-earth into the halls of Mandos:
But Lúthien came to the halls of Mandos, where are the appointed places of the Eldalië, beyond the mansions of the West upon the confines of the world. There those that wait sit in the shadow of their thought.
Lúthien sings of her fate, she will not see Beren again as he a mortal will pass out of the world.
The story eventually comes to an end when Manwe gives her the choice to become a mortal and live with Beren for another life in Middle-Earth and give up her immortality.
This doom she chose, forsaking the Blessed Realm, and putting aside all claim to kinship with those that dwell there; that thus whatever grief might lie in wait, the fates of Beren and Lúthien might be joined, and their paths lead together beyond the confines of the world. So it was that alone of the Eldalië she has died indeed, and left the world long ago.
So we can see quite clearly that the Sindar are treated the same as the other elves in Arda and are not denied immortality.
For reference, here is an image charting the sundering of the elves which includes both Círdan and Thingol (Elwë) under the branch of the Sindar.
Another of the Sindar are Thranduil and Legolas. We know Legolas sails to the undying lands sometime in the fourth age.