Sister Jeanne asks:

Is it possible that the belief in acheiropoieta of Christ began with the discovery of an image on our Lord’s burial shroud? I had learned that the idea began with the legend of the Camuliana icon. I also wonder if there are examples from other world cultures that could have led to early Christian belief in such things.

For starters there is an entry in Wikipedia (which offers several references not included here):

The image of Christ that appears in Camuliana is mentioned in the early 6th century by Zacharias Rhetor, his account surviving in a fragmentary Syriac version, and is probably the earliest image to be said to be a miraculous imprint on cloth in the style of the Veil of Veronica (a much later legend) or Shroud of Turin. In the version recorded in Zacharias’s chronicle, a pagan lady called Hypatia was undergoing Christian instruction, and asking her instructor "How can I worship him, when He is not visible, and I cannot see Him?". She later found in her garden a painted image of Christ floating on water. When placed inside her head-dress for safekeeping it then created a second image onto the cloth, and then a third was painted. Hypatia duly converted and founded a church for the version of the image that remained in Camuliana. In the reign of Justinian I (527-565) the image is said to have been processed around cities in the region to protect them from barbarian attacks.[3] This account differs from others but would be the earliest if it has not suffered from iconodule additions, as may be the case.[4]

One of the images (if there was more than one) probably arrived in Constantinople in 574,[5] and is assumed to be the image of Christ used as a palladium in subsequent decades, being paraded before the troops before battles by Philippikos, Priscus andHeraclius, and in the Avar Siege of Constantinople in 626, and praised as the cause of victory in poetry by George Pisida, again very early mentions of this use of icons.[6] It was probably destroyed during the Byzantine Iconoclasm,[7] after which mentions of an existing image cease (however Heinrich Pfeiffer identifies it with the Veil of Veronicaand Manoppello Image [8]), and in later centuries its place was taken by the Image of Edessa, which apparently arrived in Constantinople in 944, and icons of the Theotokossuch as the Hodegetria. The Image of Edessa was very probably later, but had what apparently seemed to the Byzantines an even more impressive provenance, as it was thought to have been an authentic non-miraculous portrait painted from the life during the lifetime of Jesus.

I have not seen any mention of non-Christian acheiropoieta. That is an interesting and important question.