As a socially awkward person, I understand their difficulty with being present and not speaking. The only advice I can offer is to learn to be comfortable with that discomfort, not very helpful I know. Most people are accepting of quiet people just hanging out and occasionally contributing, there's no need to hide.
Alternate input is tricky especially when we're talking about a multiplayer experience like Poker Stars VR. If they did allow you to interact solely with button pushes, what would the other participants see? Some default set of animations? A big part of the experience of that game is the social interactions around the table. Turning to speak to the person next to you, playing with all of the props provided in game, telling jokes or stories, etc. It's like a real poker game. The author seems to want a 2d experience instead so my question is why are they even bothering with VR? Why not just buy a headset that is designed as a heads up display and play regular pancake content?