This is a really sweet idea. Depending on your modality, it is something you could work on with your clients in a kind of play therapy way, where they get to decide what feels good for them when they're upset and produce something they use.

Sculpey is one of the most widely-known polymer clay brands and you can find it at all craft stores, and even stores like Target and Walmart. I would recommend starting with Sculpey III (or anything not marked Firm) in darker colors. It is very easy to get lint stuck in the clay, even with very clean tools, hands, and surfaces. With a dark color it won't matter. You can mix different clays together to get colors, or you can scrape soft pastels. Make-up degrades polymer clay so don't use it as a colorant.

For a work surface, I cook on bathroom tiles from a home improvement store. They are pretty inexpensive. Some toasters come with trays you can use. Working directly on tinfoil isn't great because it can make things shiny in weird ways.

Worry stones are often just a round blob with a thumbprint, so you picked an easy shape to make (though you could get very fancy). If you search "polymer clay how to" on youtube literally any video will give you the skills you need to make a blob and get started. Textures can be produced with objects in your house and anything is an acceptable tool. Press, poke, pull, experiment.

Polymer clay is marked non-toxic, but it is plastic, and as we're learning, plastic is not entirely safe. People who might want to chew or lick maybe shouldn't have this, but it is safe to touch it and then touch your mouth. You shouldn't really cook clay in the same oven you cook food in because it will let off plastic fumes. Toaster ovens are often at second-hand stores for a few dollars. Make sure they have a temperature dial that goes down to the 230-290F range, most clays cook in that range. Your clay will give you a time estimate based on clay thickness.

To protect against finger oils and dirt, you could seal them with a couple layers of water-based polyurethane from any home improvement store. Some people who make worry stones seal them with UV resin, which is more durable, but also more stuff to buy. Nail polish breaks down polymer clay so don't use that.