My family and I actually play league together. We constantly have issues inviting each other to games and actually being successful in starting them. With an exception to myself(for an unknown reason) everyone constantly gets 'disconnected' from league. We don't understand why or how. Is this a bug? Or just something not allowing multiple people to play together being using the same internet?

IF you were trying to do this recently league has been having pvp issues and dcing lots of people. If you've been doing this for awhile, it's your own internet.

I doubt it's the internet because everything besides league seems to work fine. It's been happening for about a month now. Issues with staying connected to game room, or Q, or in champion selection. As well as trouble inviting sometimes.