Several things to clear up.
You can’t purchase the handgun. No FFL can sell to you.
You and he live in Washington State.
As for transfer and needing background check:
(4) This section does not apply to:
(a) A transfer between immediate family members, which for this subsection shall be limited to spouses, domestic partners, parents, parents-in-law, children, siblings, siblings-in-law, grandparents, grandchildren, nieces, nephews, first cousins, aunts, and uncles, that is a bona fide gift or loan;
https://app.leg.wa.gov/RCW/default.aspx?cite=9.41.113As for possession in Washington State:
RCW 9.41.240
Possession of pistol or semiautomatic assault rifle by person from eighteen to twenty-one.
(1) A person under twenty-one years of age may not purchase a pistol or semiautomatic assault rifle, and except as otherwise provided in this chapter, no person may sell or transfer a semiautomatic assault rifle to a person under twenty-one years of age.
(2) Unless an exception under RCW 9.41.042, 9.41.050, or 9.41.060 applies, a person at least eighteen years of age, but less than twenty-one years of age, may possess a pistol only:
(a) In the person's place of abode;
(b) At the person's fixed place of business; or
(c) On real property under his or her control.
(3) Except in the places and situations
https://app.leg.wa.gov/rcw/default.aspx?cite=9.41.240So fundamentally and this isn’t legal advice but from stated RCW.
Appears you are good to go.
Buying ammo from store may become problematic for you.