I bough a tablecloth with a plastic grommet around the umbrella hole. It came loose, can I put it back?

Yes, it is very easy to put it back. Here is what to do:
First split the grommet in half with a screwdriver or other flat object using the notch on the lower half. Then place the grommet halves over the hole, placing the one with the prongs at the bottom and the other on top. Snap the two halves together using the palm of your hand.