Assume there is a willing transmuter wizard, of high enough level to make the "transmuter's stone". A simulacrum of this wizard is created.
Can the simulacrum spend 8 hours to create its own transmuter's stone as per the class feature?
Simulacrum says (emphasis mine):
The duplicate is a creature, partially real and formed from ice or snow, and it can take actions and otherwise be affected as a normal creature. It appears to be the same as the original, but it has half the creature's hit point maximum and is formed without any equipment. Otherwise, the illusion uses all the statistics of the creature it duplicates.
The simulacrum uses all the statistics of the creature it duplicates. If the ability to create a Transmuter's Stone is part of the original creature's statistics, then the simulacrum can use it, too.
The simulacrum is incapable of advancing levels or recovering spell slots, it requires "repairs" when damaged, and melts into slush when "killed". Other than that, there are no explicit restrictions on which abilities it can use. Think of it as a snapshot at the instant it is created. It can't improve itself beyond that snapshot, but that doesn't mean it can't affect the world around it.