I would look for untreated wool carpeting. If you spill something on it (vomit/pee) you can take it outside, hose it down, let it dry in the sun and put it back in place. All natural and easy care.

For small spaces, sheepskin rugs are great.

Find a carpet remnant that is nylon. It will wear like iron. This rug will become a hazmat area within a few years. A natural fiber will be a total disaster.

I'd look at a braided cotton rug. You don't say what size you want, but if it's small enough it would fit in front-loading washer.

Earthweave carpet sells rugs, but they are not inexpensive. For a plush, nontoxic, high quality rug, you are really going to pay for it. I have tried wool rugs from pottery barn kids (not cheap), and they are plush and soft with minimal odor/offgasing. They do shed, however. I think braided cotton is a great suggestion

Inexpensive? Push? Wool, No shedding or fuzzing?

You need a quality BROADLOOM store, who sells "green" wool remnants. Have it cut and bound. Otherwise? You are in dreamland with the baby.

Or settle for cute and small and washable @Lorena Canals.com

Actually, try this:

https://hookandloom.com/product/natural-wool-loom-hooked-rug-solid-natural/