Yes you can send gift cards through the mail.
When sending gift cards through the mail you should place them in a security type envelope that is not see through so that someone cannot see that there's a gift card in the envelope and try to steal it.
My aunt sends me gift cards through the mail all the time for Christmas, Birthdays etc and I've always gotten the gift cards in the mail just fine.
You can also send gift cards through email as well and the person will get it instantly when they check their email after you pay for it.
You can buy the gift card online and then send it through email which is a safer option and faster but you can send gift cards through the mail.