I did this previously by using MultiMC (now PrismLauncher). Make a mix of mods, send them the zip of the whole game instance and ask them to drag and drop it onto MultiMC. The biggest issue I encountered was one family member having a black ingame world until they changed a setting in the graphics mod, otherwise it didn’t seem to be too hard.

By comparison Minetest is much easier for playing mods with family. Everyone downloads the server’s mods when they join. But the interest is lower.