There is no rule against it.
Here are the only rules regarding wearing stuff specific to pitchers: 6-2-1
"f. wearing any items on the hands, wrists or arms that may be distracting to the batter;
g. wearing or placing tape, bandages or other foreign material (other than rosin) on the fingers or palm of his pitching hand that could come in contact with the ball;
h. wearing a glove/mitt that includes the colors white or gray;"
This is also a rule against jewelry: 2-1-12
"Jewelry shall not be worn (See 3-3-1d) except for religious or medical medals. A religious medal must be taped and worn under the uniform. A medical alert must be taped and may be visible."
Of course sunglasses are evidently not jewelry since many fielders use them.
I have seen an umpire demand that a pitcher not wear sunglasses during a FED game, but he didn't really have rule support for it. I do recommend that your son only wear non-reflective and plain sunglasses to avoid the appearance of trying to upset the batters. Lots of people believe there is a rule against pitchers wearing sunglasses, and there's no point in inflaming them.