The gasket on the rim looks like it might be "Oogoo". There was an article in American Woodturner a couple years ago about Oogoo although the article misspelled it as "Oogroo". The article featured making a gasket for a vacuum chuck. Oogoo is a DIY moldable material made by mixing silicone caulk and cornstarch. The silicone caulk needs to be the cheap stuff that stinks like vinegar (Ace Hardware brand for example) and NOT the fancy odorless caulk (GE Silicone II, for example).
Finding the "right" gasket material for a vacuum chuck is a never ending struggle between firm, soft, thick, and thin. And, the requirement varies with each piece that you turn. If the gasket is soft and thick, it is great for getting a good seal, but the disadvantage is that the turning will squirm out of alignment as you apply vacuum. A thin and firm gasket is great for bowls that don't have any warp and don't squirm much as you apply vacuum, but many bowls develop some slight warping even if they were dry when you turned them. I've found a door seal that has a D profile that seems to be a good compromise between conflicting requirements.