The real canon answer is, We don't know as of yet much about the Valyrian reserves Citadel have. So it is impossible to provide a definite answer as of now. We can however make informed speculation based on what George has provided us for.
It can be argued that the Valyrian steel is not lying around. It appears that there are pre-existing links at the Citadel which are under the custody of the Archmaesters. When a Maester learns enough about Higher Mysteries, he may request the Archmaesters to grant him one of the link.
Q: Now the question. Do maesters fully forge the links of their chokers from raw metal, or do they take strips of existing metal and
forge it into the links?
GRRM: When he said a maester "forges" his chain, it is more metaphorical. They do study metals, but that doesn't necessarily
include training as blacksmiths.
Q: I had assumed the former, but the latter would be simpler. I ask, of course, because of the link of Valyrian steel in Maester
Luwin's chain. Did he know the spells, or did he take an existing
strip and beat it into shape?
GRRM: He studied magical tomes and histories, and mastered enough of the arcane lore to impress an archmaester into granting him
a link. He did not necessarily have to make the steel himself.
The Citadel - So Spake Martin: MAESTER CHAINS AND VALYRIAN STEEL LINKS
But there can be varying interpretations. It is possible that Archmaesters could commission a smith to create a new link for the grant, from some deposit of Valyrian steel at the Citdael.
As you noted, the steel itself is very rare and the art of creating more has been lost. Older steel can be reworked but no one has been able to forge new since fall of Valyria. Also, thanks to Edlothiad, see what George says about forging Valyrian steel swords:
Q: A brief question about Valyrian steel - is it the metal that makes
the sword so special (provenance, age, etc), or is it the forging
GRRM: Forging techniques and spells, actually. There is magic
involved in the making of Valyrian steel.
Q: In other words, if a smith knew how, could he take some fresh,
high-quality steel and produce an equally fine sword?
GRRM: Not unless he could work the magic.
Q: Or is it something like Aragorn's Anduril in the Lord of the
Rings, which could be re-forged but not made from scratch, or so it
seemed to me.
GRRM: That's pretty much been the situation since the Doom of
The Citadel - So Spake Martin: PRODUCING VALYRIAN STEEL
Q: I hope all goes well with you and this finds you in good health. I
was just wondering if you could settle something for me in relation to
Valyrian steel and I I [sic] just wondering if I have taken you up
correctly on the matter. I belive I'm right in saying it differs
from say Mithril (from LOTR) in that it is not a material which in
itself bears advantageous properties but rather ordinary steel which
has been subjected to a process (the physical manipulation of the
steel combined with spells) which embues it with the desired elements.
GRRM: Yes, that is correct. You don't mine Valyrian steel
(actually, you don't mine any steel), you make it.
Q: What I'm less sure of is whether Valyrian steel ever exists as a
GRRM: It does not.
Q: I believe it doesn't but only as a finished blade, what I mean is
that it is the actual process of making the sword from run of the mill
steel which gives us a Valyrian weapon rather than Valyrian steel
being made beforehand and then this product being used to make an
GRRM: The closest real life analog is Damascus steel, but Valyrian
steel is a fantasy metal. Which means it has magical characteristics,
and magic plays a role in its forging.
The Citadel - So Spake Martin: THE PROCESS OF MAKING VALYRIAN STEEL
So it seems clear that the art of creating new Valyrian Steel weapons from the scratch has been lost since the Fall of Valyria but old ones can be reworked.
Given that, The whole process of the link-award is feasible only if the links return to the Citadel on the awardee Maester's death or expulsion. Given that Maesters do not surrender their chains to anyone while they live (unless expelled by the order) and Samwell took Aemon's chain back to the Citadel, that seems likely.
The captain wanted Aemon’s chain as well, but there Sam had refused. It was a great shame for any maester to surrender his chain, he had
explained. Xhondo had to go over that part three times before Quhuru
Mo accepted it. By the time the dealing was done, Sam was down to his
boots and blacks and smallclothes, and the broken horn Jon Snow had
found on the Fist of First Men. I had no choice, he told himself.
AFFC - Chapter 35
But that is my own speculation, there's nothing in canon to suggest that it works that way. We will know soon enough, we have just started learning about the Citadel after all. Sam's sojourn there will teach us a lot about their way of life.
Furthermore, Why would Citadel hand over their rare links to other people to make weapons? It's their property. We don't even know how many links do the Citadel have? Do they even have enough to make a sword? We only know two Maesters who have those links. If they had enough links, why didn't any Targaryen monarch ask them to hand them over when Blackfyre was lost? Why didn't the Lannisters make some ludicrous offer, specially as Tywin is known to have made generous offers to impoverished houses with Valyrian swords?
So in conclusion:
- There might not be "unused" Valyrian steel at the Citadel.
- Even if there is, New Valyrian Steel weapons cannot be created as the art is lost so there is really no point in procuring the links from the Citadel.
- The Maesters do not work the steel.
- The links presumably return to the Citadel on Maester's demise or expulsion.
- There is no indication if there are even enough links at the Citadel to forge a longsword. We only know two Maesters who have such a link, Marwyn and Luwin.
- The Citadel would have to be utter fools to sell the rare resources to bidders. They like to horde their treasures.
- Given the continued patronage and protection of mighty Hightowers, They can easily defy other Lords who might demand them to hand over the links.