Let me try and explain this:
The avidness with which merchants and landowners in early-nineteenth-century Maryland and Virginia sought Joshua Johnston’s professional services attests to his artistic skill as a portrait painter. What does this sentence mean? What does the “which” indicate here?
In the early nineteenth century merchants and landowners from Maryland and Virginia wanted to use J J’s professional services. The fact that (because) they tried to get these services with so much enthusiasm/determination (avidness) proves (attests to) that he was a very skilful (clever) painter.
Hope this helps.