Yes palmetto bugs can make you sick and they do carry diseases.
Palmetto bugs are the same thing as cockroaches and are the American version of the cockroach.
When the Palmetto bug or cockroach gets into your food and contaminates your food then when you eat that food that was contaminated by the palmetto bug or cockroach you can become sick from it.
If cockroaches or palmetto bugs have been into your food you should throw the food out to avoid being sick.
Also the species of cockroach palmetto bug emits a chemical spray that they can spray onto your dishes, utensils etc which can make you sick as well.
You may not even know that the roach sprayed your dishes or anything else with the chemical spray because you can't really smell it.
If you have any palmetto bugs or any other species of cockroaches you need to get rid of them fast.
I would recommend either fogging the house with roach fog or set out some of those sticky roach traps or roach motels or hire an exterminator or use some roach spray around your house to kill the roaches.