I do not know when crochet kits were first sold. The earliest ones I have seen are dated from the 1930s.
Pictured here are some from my collection. Starting at the top left corner and going clockwise, ending with the kit in the center, they are:
Susan’s Crochet Doll Kit, G.A. Doherty Co., Buffalo 3, NY. (no zip code, so pre-1963, looks 1940’s-1950’s to me) contains DMC Cotton thread and plastic doll; hook no longer in my kit.
Vogart Crafts Christmas Crochet kit, NYC address. Dated after 1973 when Vogart Co. became Vogart Crafts Corp., but prior to 1981 when they relocated to Clifton, NJ. Illustrations on side looks mid-late 1970s “Crochet hook and Polyester stuffing included.”
Pierre the Poodle. Copyright 1961. “All materials included to make Pierre” including synthetic yarn. My used set does not have a crochet hook; contents displayed in separate picture.
Clark’s Crochet Package ~1930’s photos. Contains Clark’s cotton thread and instructions to make 1 of 4 collars. No crochet hook.
Lady Alice’s Easter Lily Corsage. Lady Alice Acrylic yarn dated 1977. No crochet hook.
Alice Sophie Wallace The Happy Surprise Tulip Bouquet. Acrylic yarn dated 1978. No crochet hook.
Wonder Art (Fixlers Bros.) of 3900 N. Claremont Ave., Chicago, ILL. Contain-All Crochet Kit. Moved to the cited address in 1931; no zip code so before 1963, looks 1930’s. Contains Heirloom Crochet Cotton. Does not have a crochet hook, but may have been removed.