There are indications of a money economy, but Evermeet appears to be self-sufficient.
The 5e Sword Coast Adventurer's Guide appears to be silent about this topic. So here are some lore from earlier editions.
There is a 2e sourcebook, Elves of Evermeet. In it, money is not explicitly discussed, but we can infer the presence of some form of trade and economy of goods from the description of various locations. For example, in the description of the settlement (capital city) of Leuthilspar, we read about:
- Tamnaeuth, the armorer, who does not charge money for his crafts, but selects his own clients. [This implies to me that others would charge.]
- Thaola's Wineshop, selling fine elven vintages.
- Shael, the clothier, who charges very little for his items.
- Sunstone Commons, an open area where elven merchants sell their wares.
The 3e Forgotten Realms Campaign Set lists the following about Evermeet's economy:
- Imports: Elven items retrieved from Faerûn, otherwise self-sufficient
- Exports: Elven adventurers, jewelry, some exotic or magic items given as gifts
All these data come from a time when Evermeet was entirely on Toril and not in between the prime material plane and Feywild. It is up to the DM to rule on the current status.
If you think unofficial sources are ok, there is also a Candlekeep discussion (21 Mar 2019) on buying a home in Evermeet or Evereska, which provides some clue on the topic. A post states:
Ed recently said that elves don't really deal much in gold when dealing with each other; but in rare seeds, plants and so on.
Unfortunately, I have failed to identify the source of the original statement by Ed Greenwood.