That's not a stupid question. When I googled it was in an article about the shroud,
In the year 1969, on the direction of Cardinal Pellegrino of Turin the Shroud was subjected to further analysis by an international panel of scientists, that included a syndonologist to study the texture of the cloth.
So, we know they study the texture of cloth, beyond that I don't know.
I think sometimes it is written as Sindonologist. Which would be someone who studies the Shroud (of Turin I assume. What other Shroud would someone study???)
So, in conclusion, I still have no idea what syndonologist does.