Is there a name for a place that sells nuts?

The word nuttery is defined as:

: a place where nut trees grow; also : a place for storing nuts.

So, a "store" for nuts is a nuttery.

"Roastery"
"Nuts & Coffee Roastery"
Where brewed coffee is not served, it is only for selling grounded or whole grains coffee beans and roasted or raw nuts

There isn't a widely recognized term like butcher or pharmacy, but nut shop would describe the store and nut seller the shopkeeper. The Oxford English Dictionary has the latter listed under "nut, n.1 and adj.2," referring to the person (the monger) selling the nuts:

nut seller n.

1648 H. Hexham Groot Woorden-boeck Een note-menger, a Nut-seller.

1851 H. Mayhew London Labour I. 201/1 These almond nut-sellers are, for the most part, itinerant.

1990 Chicago Tribune (Nexis) 25 June (Business section) 5 The tentmakers share a street, as do the nut sellers, the gold merchants, and those selling silk or coffee or discount underwear.

Nut shop does not have a similar entry, but the same pattern exists: the nut shop is where nuts are sold. Some results also point to its use:

• the above image and many others can be found in Alamy under "nut shop,"
• Now-closed New York City nut shop "We Are Nuts About Nuts" was called a "nut shop" by both an Eater New York article ("by the time the nut shop opened, the area had a sexy, new TriBeCa moniker") and interviewee David Dyssegaard Kallick: "I like that nut shop. I’m sorry to see it going. There’s some loss to the character of the neighborhood without it."
• The Chicago Tribune reported on another place opening last year: "Octogenarian and her daughter open nut shop in Highland Park amid pandemic." In the article, one of the founders says: “But I think that people are really looking for something positive and even if it’s just this tiny little nut shop, they just want something to feel good about.”