There does not seem to be a standard notation for spoons; however, spoons are related to castanets, and there are some examples of castanet notation, though also not standardized.
Castanet notation #1
The book Castanets: Study of Rhythm Music, Book 1, by Emma Maleras, uses a two-line notation. The upper and lower lines denote left and right hands, respectively, and simultaneous strikes are indicated by notes on both lines, or by notes in the center space, depending on the type of stroke to be used.1
Castanet notation #2
Tabourot's book Castañuelas, Olé!: A Book about Castanets, 2nd ed. gives a notation attibuted to "Monsieur Feuillet". This notation is similar to the Maleras notation but uses a single staff line with right hand notated above and left hand notated below. The type of stroke is given below the notes in the position might find lyrics in a vocal score.2
Castanet notation #3
In an arrangement of Boccherini's "Fandango", the castanets (Percussion 2) are notated on a single staff line designating rhythm only, leaving it to the performer to decide on the technique.3
Toward a spoon notation
It does not seem a stretch that a spoon notation could be adapted from these, possible also incorporating up and down stroke symbols from guitar notation. Techniques or strokes on different parts of the body could be notated directly through text.
1Maleras, Emma. Castanets: Study of Rhythm Music, Book 1 (Editorial Boileau, n.d.). A preview of the book contains the information found in this post.
2Tabourot. Castañuelas, Olé!: A Book about Castanets, 2nd ed. (Tactus Press, 2007).
3Boccherini, Luigi. Guitar Quintet in D major, G.448, "Fandango" from Mvmt. 3, arranged for orchestra by Charles Anthony Johnson. (IMSLP)