Is there a specific word for friends who you play cards or board games with regularly?

The following is understood and grammatical to this AmE ear"

Sometimes I cover for one of my grandma's regulars and play Mahjong with them.

The OED supports the sense of such usage: a regular

noun - a person who does something or goes somewhere on a regular basis; a regular or habitual customer, visitor, participant, employee, etc. Now esp.: a person who habitually frequents a particular bar, restaurant, etc.

As in:

Glancing over the crowd, he saw many of his regulars.

and

This gentleman was one of his regulars.

I think you have two problems:

1. how to refer to the people that meet regularly
2. how to refer to someone within that group of people

Problem (1) can be solved by referring to the group as a "Mahjong group". It implies all the requisite information - Mahjong is a game, so members are assumed to play, and "group" implies regular attendance. Problem (2) is then trivial, because any member, player, lady, man, someone in that group is automatically an habitual player.

Sometimes I cover for a player in my grandma's Mahjong group.