A vinyl tile doesn't really need a finish and because it is designed to reject dirt, might repel a finish that is put on it.
That being said, the best way to find out is to try a finish on a loose tile and then abuse that tile in whatever manner you imagine your finished table might be used at its harshest jobs. Try one sanded a bit to grab the finish a little if that doesn't mess it up too much.
I would find a finish with low VOC's that complies with the South Coast Air Quality Management Board, one of the strictest in the country. You can call the manufacturer listed on the can to ask for this information. A clear finish might not be absolutely zero VOC and non toxic as that is harder to come by than a pigmented paint.
Besides non-toxicity, you'll want durability too, of course. Because we don't know the make up of the tile you are using, this is impossible to address. Not all plastics and vinyl are equal when dealing with finishes and sealers.
You are going to have to dive in here and experiment.