The correct option is
A his/himself
The given sentence speaks of Peter.
A possessive adjective is usually used to describe a noun, and it comes before it, like other adjectives.
So, the first pronoun of the given sentence must be an adjective pronoun. Since the subject is male, the adjective pronoun 'his' must be used.
We can thus eliminate options A and E as incorrect options.
When the subject and the object of the verb is the same person or thing, we use a reflexive pronoun. Here,Peter is both the subject and the object of the verb.
The reflexive pronoun for singular third person is 'himself','herself' or 'itself' according to the gender of the noun.
In the given sentence the noun is a male. So, the reflexive pronoun used must be 'himself'.
Hence the complete sentence must be-
On his way home Peter decided to buy new skates for himself.
Thus option D is the correct answer.
Options A,B,C,E are incorrect.