It is quite probable that your general dentist doesn't have a papoose board, at least in an adult size. Papoose boards are kind of rare these days, because they are, in essence, restraining devices, and those techniques have gone out of fashion.
I can see how a papoose board would make you feel safe and secure. On the other hand, it puts certain restrictions on you. For example, you can't give stop signals to your dentist whenever you want to take a break.
Do you reckon a heavy blanket could fulfill the same purpose? Some people just like the feeling of security they feel when "protected" by a blanket-like device. If so, you could bring your own. Of course, you could still ask if your dentist has a papoose board for adults, but probably they don't.