There were always coins with portraits on them, even if most of the early depictions are rather crude by our standards. Though putting portraits on coins was obviously done much earlier than the Middle Ages, as many Roman coins had portraits of living people on them, since you specifically addressed the medieval time period, here's an example of a coin from the 9th century with Alfred the Great's face on it.

I don't know about the laity's access to art outside of the church, but royalty would often travel from town to town to assert their public authority. Cities would hold lavish processions and festivals known loosely as "royal entries." They were often symbolically and visually linked to Christ's entry into Jerusalem. Sarah Beckwith talks a bit about this in "Medieval Penance, Reformation Repentance, and Measure for Measure" which you can find in Reading the Medieval in Early Modern England

My knowledge of this mostly comes from reading about the York Mystery Plays (English Lit person), so someone with more expertise might be able to be more thorough.

An odd example... Various churches in 14th-century Padua bore the visages of Carraresi nobles surreptitiously worked into frescos. Petrarch appears in a few as well. The question is whether these portraits would have been immediately recognizable to the average merchant or tradesman attending mass. But note also that churches often served as hideouts for runaways, prostitutes, petty thieves, etc. (blanking on a source for this), so such portraits would have reached a large and varied audience.

Source: Diana Norman, ‘“Splendid models and examples from the past’: Carrara patronage of art,’ in Siena, Florence and Padua: Art, society and religion 1280-1400 (Milton Keynes: 1996), 159