I put dish detergent on the spots and ran water through the fabric when one of my cats "blessed" one of my projects.
If it's fresh, most of it should come out.
Baking soda is another thing you could try.
I would try this on a scrap first, though, to see if or how it affects the colors/fabrics.
I would also avoid "scrubbing" or "agitating" as much as possible.
One of my cats throws up a lot because he has hyperthyroidism - even with medication - I think about half of what he eats comes back up. I'm reasonably sure he can't help it.