You do need to buy the game (once) off course and you cannot play online.
Download the 'old' installer from: https://launcher.mojang.com/download/MinecraftInstaller.msi
Do not use the lastest installer and never upgrade to the new minecraft launcher when asked.
Install the game, login using your Microsoft account and start up the game.
Repeat this process on all PC's that are going to join the LAN game.
When two people join one server using the same name, one will be kicked. When you launch Minecraft, your player name will be loaded from your Microsoft game-profile. So how can you change your name I hear you ask?
You'll need to repeat these easy steps each time you want to play on a LAN game.
You can see the name change in the launcher:
Also, it's worth setting up a dedicated Minecraft server for your kids. I've setup a survival server (with monsters and no cheats) and a creative server with unlimited life, flying and all resources.
More on hosting your own dedicated lan server here: https://help.minecraft.net/hc/en-us/articles/360058525452-How-to-Setup-a-Minecraft-Java-Edition-Server
It's super easy and doesn't require a fast machine.