No, any white (regular) Storylets that you are eligible for shouldn't disappear, but you may eventually become ineligible for them.
Like you've said, while Storylets of other colours may disappear after they are completed, white Storylets normally don't.
Here is an official source on the matter (taken from the "What are Storylets?" section of the in-game help):
All Storylets that the player is eligible for in a given location are always available.
Now, what may eventually happen is that you find yourself ineligible for old Storylets. While most Storylets have a minimum Quality level requirement, some have a maximum level requirement. If your Qualities exceed a maximum level requirement, the Storylet will be locked.
Here is an example of a Storylet that has both a minimum and maximum Quality requirement (which I have exceeded):
What can be even more confusing, however, is that while some Storylets may have maximum Quality requirements, they aren't explicitly told to you.
Here is an example of a Storylet that only says "Unlocked with Shadowy 32", but that has been found to be no longer available to players with Shadowy 47 and above (See "Rob a honey den" in the Echo Bazaar wiki):
I've only really seen this happen with Storylets that take place in Veilgarden, Ladybones Road, Spite, and Watchmaker's Hill and only with Storylets that just require one of the four major Qualities (although I can't be 100% sure).
While some might find this annoying, it seems to be done as an effort to keep these locations from becoming cluttered with Storylets that are largely irrelevant to higher level players.