The sun was made from the last fruit of Laurelin (one of the two great trees) and is being shipped around the sky on a cart. Legolas has never seen the light of Valinor, i believe he was born after the trees died. It’s definitely not visible at all times to the elves, since even before the world was turned round it went below it out of sight at night-time. Presumably the elves still see the sun affect only parts of the world, they can just see beyond the horizon.

Edit: as an aside, the last remaining silmaril is the Evening star that shines in the west.