Can we use one Minecraft account to join a LAN game with multiple players?

TLDR: Yes!

You do need to buy the game (once) off course and you cannot play online.

How then?

1. Prepare each game rig

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.

2. Change Player names

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.

1. Don not start the game yet!
2. Turn off the internet connection on your machine (by using Airplane mode).
3. Open Explorer and go to this directory: C:\Users\{your username goes here}\AppData\Roaming\.minecraft\
4. Open the file launcher_accounts.json in any text editor
• Optional: Create a shortcut to this file on your desktop for easy access.
5.  Change the player name in the JSON file under "accounts > mineCraftProfile > name" and save the changes
   
   {

     "accounts" : {

       "ece1f9ba06666666666666a92154c4" : {

         "accessToken" : "",

         "accessTokenExpiresAt" : "2022-02-20T14:05:48Z",

         "avatar" : "data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEAAAABACAY...",

         "eligibleForMigration" : false,

         "hasMultipleProfiles" : false,

         "inForcedMigration" : false,

         "legacy" : false,

         "licenseProductIds" : [],

         "localId" : "ece1f9ba66666666666666666a92154c4",

         "minecraftProfile" : {

           "id" : "1116e34c6666666666666662e18ba30",

           "name" : "Player name goes here"

         },

         "persistent" : true,

         "remoteId" : "2535446666662750",

         "type" : "Xbox",

         "userProperites" : [],

         "username" : "Leave this name! this is your microsoft account name"

       }

     },

     "activeAccountLocalId" : "ece1f9ba066666666666662154c4",

     "mojangClientToken" : ""

   }
6. Start the Minecraft Launcher
7. ONLY now turn your Internet connection back on.
8. In the Mincraft Launcher choose, PLAY OFFLINE. (If you see "demo", you need to turn on internet and login with the microsoft account again).
9. Join the server & enjoy the game!

You can see the name change in the launcher:

gl&fh

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.