Laity is a residual category. It means everyone who is not clergy, and it therefore cannot exist where clergy do not.
That absence is not a gap in those traditions. It is a structural claim they are making.
The Religious Society of Friends has no clergy, no ordination and no minister. A meeting for worship is held in silence and anyone present may speak if moved to; nobody presides. Business is conducted in the same manner, with no vote taken, the clerk discerning the sense of the meeting rather than counting. Friends do speak of someone having a gift in the ministry, but this is recognition of something already happening, not appointment to an office, and it confers no power over anyone. The arrangement is a deliberate seventeenth-century judgement that a paid separated clergy is an obstacle rather than a help, and it has been maintained for three and a half centuries.
The Plymouth Brethren made a comparable judgement independently. So, in different terms, did a range of radical Reformation groups.
Islam's self-description is that it has no priesthood, and while the scholarly class exercises great authority, no rite creates it and no function is reserved to it.
Most Hindu practice does not run through a clerical class either. Temple priests exist and perform specific rituals, and domestic worship, pilgrimage, meditation and the reading of scripture do not require them.
Where the distinction does exist it does heavy work, determining who may preside, who teaches with authority, who votes in which body, and who may hold office. The Second Vatican Council substantially raised the theological standing of the Catholic laity, describing them as sharing in Christ's priesthood and bearing the primary responsibility for the ordering of secular affairs, while leaving nearly all the governance restrictions where they were. That gap between elevated status and unchanged power is where most subsequent argument about lay participation, and about women in particular, has taken place.