Here’s your answer. NO.
Jesus was a poor man who along with his band of apostles traveled the region with nothing but the clothes on their backs. He sent apostles to find the man holding the donkey and to tell them the Master needs his room that night.
I doubt they brought along a formal table cloth.
Being poor, a hand woven item like a tablecloth would have been a massive luxury for nearly anyone except the very rich, or perhaps the Pharisees and corrupt priests of the day (pretty much all of them)
Speculation of an item like this is nuts and does not mesh with Scripture.