Keggy,
I suppose marbles are a bad idea with swallowers - sorry about that...
What about doing it on a bigger scale? a bigger board would mean being able to use bigger balls that can't be swallowed (or can't even fit in a mouth).... or you could use some sort of round candy as the rollers instead of marbles.
about the paper chips.... - what about getting a large circle punch and punching out circles from the thin foam board (I'm a big fan of foam board, can you tell?)
The foam would be heavier than paper, last longer, and with a large hole punch (can be found in the scrapbooking aisle) would be easy enough to do.
As fo adding the hospitals name to the chips - it's fairly easy to get one of those "return address label" self-inking stamps that come with letters/numbers so that each person can make it to have their name/address on it (you can definately find these at Staples) and you could use that type of stamp to make the hospitals name (or initials at least) to stamp it on the chips.