Yes, your son will be liable for any capital gains tax due when he sells the gifted stock. This FAQ explains how he will determine the stock's cost basis when he sells it: https://ttlc.intuit.com/questions/3353068-how-do-i-determine-the-cost-basis-of-stock-i-received-as-a...
Your cost basis is the market value of the shares on the day they vested to you. So when your son sells he'll need to know that number, as well as their market value on the date of the gift.
This TT FAQ explains the ins and outs of the Kiddie Tax, and should answer your question about that: https://ttlc.intuit.com/questions/1900671-what-is-the-kiddie-tax
Note too that, if the value of your gift (on the date of the gift) exceeds $15,000 (2018), you must report the gift to the IRS using Form 709. There is a lifetime exemption of $11.4 million, so it is extremely unlikely any gift tax will actually be due - but gifts over $15,000 must still be reported by the donor. https://www.irs.gov/pub/irs-pdf/f709.pdf
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
