I have bought my granddaughter a hickory ( unfinished ) youth custom bow for christmas she wants it to be ky.(wildcats) blue. Which would be the best way to go paint 0r dye ? Followed by a coat of sealer.

Dye will penetrate better than paint and will not chip off. I't may raise the grain a bit but just carefully sand with 320 or higher and then polyurethane or another finish.