I agree with Dan that the bigger safety question is the size of the puzzle pieces. Look for patterns for simple puzzles, with only 3 or 4 pieces, that are large enough not to pose a choking hazard.
As for a finish, for that age of child, I would opt for no finish. Just use a hardwood, with a tight, close grain, such as maple, and leave them unfinished. Not that the finish is a hazard. Like Dan said, all wood finishes, sold in the USA, are non-toxic, once fully cured, so you don't really have to worry about it. It's just that there really isn't much benefit to be gained by putting a finish on a puzzle for a 1 to 3 year old child. If you want to use something, I suggest shellac.