Making a family card game playable online
VASSAL was made for precisely this: it's a game engine/website where you can build games, and then play the games you built online. There are a wide variety of existing games already built for VASSAL, but Muggins is not one of them.

This will not be simple: you're going to need to learn how to program the game using VASSAL's editor. It'll take time and effort, but you will be able to re-create your family game of Muggins, house rules and all. Once you've built the game (the site calls it a "module"), VASSAL will host the game, so your family can join; you don't need to worry about the details of building a website yourself. VASSAL is free, though the site appreciates donations to keep it running.