A slice of soft white bread (crusts removed) can work well--torn into small pieces and then pureed after it has soaked up some of the stock.

You could use creme fraiche......

Of course there is soymilk, and other nut and grain milks; almond milk rice milk, etc. I agree with the above that you should try to choose a complimentary flavor profile. My favorite is coconut milk, i think its usable just about anywhere you would use milk/cream.

Remember that coconut milk also comes in various fat percentages, just like regular milk, so you could use coconut cream here for a richer taste and more body, probably the original intent of the cream. If you don't have coconut cream, you can always refrigerate a can of milk and skim the thick part from the top.

For a more neutral flavor profile, you can try adding some blended silken tofu, which is generally substitutable 1:1 for cream in dishes like this, or thin it out with one of the above mentioned milks.

What's the main component of the soup? Fish, flesh, fowl, or vegetable?

Depending on the soup, I may omit it. I like Syronai's idea of nuts, but I've heard blending cashew nuts give great results. Mollie Katzan does a carrot with cashew.

For me, that really depends on the soup. You can blend nuts (pecans, hazelnuts, walnuts) into squash or vegetable soups to make them thicker and give a creamier texture. You can add coconut milk, coconut cream, or soymilk to give flavor and richness. I pretty much stay away from dairy, so usually I'll try the soup without any cream at all and add one of the above if it seems to be really missing something. Most of the time, it's great without.