How do the non-ordained participate at Mass?

The non-ordained participate at Mass by the virtue of being present at the Liturgy. The non-ordained don't have to try to join or attach themselves somehow to Jesus. In other words, the non-ordained do not have to consciously unite themselves to Jesus during the consecration.

Just being present at Mass (Luke 10:38-42), joining in the singing, responding to the prayers, and following along attentively at Mass, the words and actions of the non-ordained are joined to Jesus, who uses their offering as a means to shower all souls with graces. It is important to mention that Christ does not require offerings (words and actions) from us to be the means for Him to shower the world with graces. Since nothing is impossible for God, Christ accepts our offerings not for His benefit, but for our own.

[For more information on the role of the non-ordained at Mass, check out the following: LPCA's Index on Vatican II's true vision for lay activity, Romans 12:1, 1 Peter 2:4-5, CCC-1368, CCC-1369, Lumen Gentium 11, Lumen Gentium 28, Paragraph 13 of ECCLESIA DE EUCHARISTIA by Pope John Paul II and Lumen Gentium 34]