3 days (or so) ago, I added a second character. How do they get the shovel recipe? I tried talking to Blathers, but he didn't offer it. Is there another way to get it, or do I just need to wait longer?
Pretty much what Jhine7 said. Nook's Cranny has certain DIYs for getting around the island. There are DIYs in the Town Plaza building when you go to redeem Nook Miles from the machine on the right. I don't know if it has tools or if that's just for more hair variety.
If I remember correctly, the character won't be automatically given a shovel DIY. I always buy it at Nook's. I know it's annoying to load your main then drop the bells assuming the alt. character doesn't have any but that's how it is.