I am assuming you are self-employed and in the business of selling animals. This means that all of your business income and associated business expenses are reported on SCH C as a physical part of your personal 1040 tax return. That is, unless your business is set up as something other than a single member LLC or sole proprietorship. So yes you are required to provide the W-9. But you are ***NOT*** required to provide them your SSN and I recommend you never under any circumstances do so. Instead, provide them your business EIN. If you don't yet have an EIN for your business then you can get one in about 10 minutes and it's absolutely free. Just go to https://www.irs.gov/businesses/small-businesses-self-employed/apply-for-an-employer-identification-n... and work it through.
You can get a blank IRS Form W-9 at https://www.irs.gov/pub/irs-pdf/fw9.pdf
FInally, if this particular customer of yours has paid you *LESS* that $600 during the entire tax year, then they are not required to issue you a 1099-MISC and have absolutely no legal need for your business EIN. But there's really nothing wrong with them requesting it, or you providing it to them. For amounts less than $600 while the 1099-MISC is not required, it is still optional. When they issue you the 1099-MISC the income they paid to you for the entire tax year should be reported in box 7. But if it's in box 3 that's fine too.
