Yes, but to what end? Does Vash become a member of the crew somehow? And what logic would there be in her returning frequently or that she would keep crossing paths with Picard? The show isn't about Picard and his crew as relic hunters or archeologists. You can pigeonhole the series like that, what's great about Trek is how it can switch it up each week and you never know what kind of episode you're going to get - mystery, Western, action, political, comedy, etc. Vash is a character with a very specific purpose and i don't see that specificity working on a constant or frequent basis on the show.
Actually, TNG did do numerous stories that in someway dealt with archaeology. Years ago while doing research for a Whoa Nellie story, I realized that for someone professing to love archaeology Picard destroys quite a few archaeological treasures in series canon – the Tox Uthat, the Stone of Gol, an Iconian gateway and a Promellian battle cruiser. In several of my stories, I have Vash refer to this as his Litany of Archaeological Sins.
Episodes that dealt with archaeology or long dead ancient civilizations (that I could think of I'm sure I missed a few):
Season 1
The Last Outpost
The Arsenal of Freedom
Season 2
Contagion
Season 3
The Bonding (Lieutenant Marla Aster was an archaeologist killed during an away mission)
Booby Trap
Captain's Holiday
Season 4
Devil's Due (dealt with the civilizations ancient history)
Qpid (archeology conference)
Season 6
Relics (Ancient Dyson Sphere)
The Chase
Season 7
The Gambit I
The Gambit II
Masks
And this would've evolved very naturally, because every civilisation has a history. Every planet's Antiquities are the object of a collector's avarice. She might even become a temporary "special advisor" for ENTERPRISE, in exchange for transport. Particularly if her knowledge about something would be key to understanding an episode, or two.

It's not uncommon for science fiction shows to use archaeologists as reoccurring characters – Babylon 5 and Stargate SG1 to name two. My favorite conversation between a sci-fi hero and reoccurring archaeologist character comes from the new Doctor Who. In “Silence in the Library" the tenth Doctor has this exchange with River Song when the character is first introduced:
The Doctor: Oh, you're not, are you? Tell me you're not archaeologists.
Professor River Song: Got a problem with archaeologists?
The Doctor: I'm a time traveler. I point and laugh at archaeologists.
Professor River Song: [offering handshake] Ah. Professor River Song, archaeologist.
Warmest Wishes,
Whoa Nellie
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