After you upgrade your tent to a house, you can talk to Timmy to build the shop.
Upgrade Your Tent to a House, and Build Blathers's Tent
Once your tent has been upgraded into a house, and you have built Blathers's tent, you can talk to Timmy in the Resident Services tent and ask about building a shop.
Timmy needs you to bring 30 wood, 30 softwood, 30 hardwood, and 30 iron nuggets. You can talk to residents on your island to get a stone axe. Use the axe on trees to make wood come out, and use the shovel, which you got from Blathers, to hit rocks until you have 30 iron nuggets. You can use Nook Miles Tickets to go to random islands in order to find more rocks and trees to get materials from, if you can't get enough from your island.
Build the Shop Construction Kit
After you give Timmy the materials, he gives you the Shop Construction Kit. Build it on your island where you want the shop to be. Nook's Cranny will open the next day.