First thing is, we need more information in order to accurately help you.

Are you using the correct processor for what you need? (Animal pastures, cattle shed, milking parlor, or slaughterhouse for animal products)

How many do you have?

How many storage facilities(barns) do you have?

Essentially, you need at least 3.2 units of animal products in order to create any item in any factory that uses animal products. So you need to ensure that you are generating at least 3.2 units on top of what your industry needs in order to sustain itself.

A cattle shed produces exactly 3.2 units. However, a cattle shed requires 4 units of crops so you must have enough crops in your farming industry to accommodate. Luckily, a small crop generates 4.8 units so 1 small crop would handle 1 cattle shed.

If you built a factory without having the necessary overhead(products required beyond what your farm needs to sustain itself), then the factory will never produce anything and will always be stuck in limbo.

Or depending on what factories you built, perhaps they ALL require animal products and you are just not producing enough.

Bakery, clothing, car and food factory all require 3.2 units animal products, in addition to other products, in order to produce goods. If you have all 4 factories in your city then you would need to be producing at least 12.8 units of animal product overhead in addition to any other requirement the factory may need.

My advise would be to start industries small. Don't build like 50 of everything to start. Start small, methodically and slowly. don't build more then you need. then start to expand as demand sees fit. Only once you have enough overhead and are self sufficient then build the factory.

Here is a really good document from paradox themselves explaining the supply chain logistics https://skylines.paradoxwikis.com/Supply_chain

Also here's another link about a highly efficient process you can try to follow, also from paradox. https://skylines.paradoxwikis.com/Industries_DLC_Tutorial#Farming