Spidersilk is definitely a thing, unsurprisingly associated with the drow. They use primarily aranea or drider silk for it, which is ironically probably easier than trying to get actual spiders to behave well enough to collect sufficient silk from.1 Most famous is the piwafwi, the drow version of the cloak of elvenkind, which is made from spidersilk, but spidersilk armor and the like have also been described. It typically provides armor similar to studded leather, but at vastly lower weight and with vastly increased mobility.2
The “v.3.5 revised edition” sourcebook Tome of Magic includes “shadowsilk,” but no mention of whether or how it fits into Forgotten Realms is offered in that book, nor is it discussed (to the best of my knowledge) in other books. Likewise, Races of the Wild from the same edition had “thistledown,” a lightweight but somewhat bulky, “downy” material used for padded armor or for the padded lining of heavier armors. It’s good for arcane spellcasters (though the ways in which it is good for them are irrelevant in 5e), but in any event that book doesn’t describe thistledown’s place in the Forgotten Realms, and it was never mentioned again.
In 4th Edition, there were upgraded forms of every type of armor, getting increasingly exotic, and one of the armor types was cloth. “Feyweave armor is woven with techniques perfected by the eladrin. Starweave armor is fashioned after patterns created in the divine domains of the Astral Sea.” Neither Forgotten Realms book in that edition—yeah, there were only two—mentions either specifically, but as part of the default game, it would be expected that Forgotten Realms adventurers would have access to them one way or another.
As far as I know, 5th Edition has yet to describe any exotic fabrics or textiles specifically.
And I don’t know of anything that discusses using any part of a unicorn’s body aside from its horn.
There have been modern attempts to domesticate spiders in the real world, or at the very least engineer habitats where spider silk could be collected from “wild” spiders. It hasn’t gone very well: spiders insist on too much space for themselves (and will tend to kill and eat those who encroach on it), which tends to lead to awful economies of scale. It’s bad enough that there were even attempts to genetically engineer goats that produce spider silk instead of milk—which worked, up to a point, because they could produce goo with the appropriate proteins in it, but the unique “spinneret” organs on a spider are necessary to actually weave those proteins into silk.
For example, in 3rd Edition, where every armor had a maximum Dexterity bonus like 5e medium armors, spidersilk armor offered a colossal +8 maximum. Only three armors in the game had a higher limit (thistledown, nightscale, and twist cloth), and they all offered lower innate benefits to AC, though a sufficiently-Dexterous character might still prefer them.