Well I was thinking if you thought it was outright wrong of them to ask, I would notify the sheriff that they are doing something wrong.
Even if there was some problem with it, there is a difficulty with having a county authority investigate an entity licensed by the Federal Government. I suspect the Sheriff would politely thank you for your report and you'd never hear from his office.
It's an issue of the Authority Having Jurisdiction, and they protect their administrative boundaries jealously. Plus, a Sheriff often has to stick to his "lane," or the County Commissioners, much less the public, will call him out for overstepping his bounds - and spending money on an investigation outside his responsible area.
A licensed Federal Firearm Dealer answers to the BATF, not the County Sheriff, on matters relating to firearms transfers.
As to whether your SSN could somehow be stolen off his records, it's not on a computer database that would be hacked online. FFL's don't bother, it's on the form, that's good enough. The book is put on the shelf. His inventory of guns means he has better security than most of us do at our home or work, nobody is getting in to steal it, and he was vetted by the government - meaning he had no past criminal background. Felons don't get a licensed to buy and sell guns, they are prohibited in most jurisdictions from even having one of their own.
How many gas stations used to require your SSN on your check buying gas in the past? Grocery store? In the day, your SSN WAS your license number in MO, and in a lot of other state up until a few years ago. Even the military did it for 50 years.
Only until recently have we shied away from it, because of ID theft, and frankly, it was already after the horse bolted from the barn. SSN's are on your bank account, are credit card accounts, car loans, home loans, title loans, building loans, utility accounts, ad infinitum. Hundreds of databases accumulate your SSN over the span of your lifetime every time you apply.
It's one of those asterisked "Must supply" lines on a computer form online.
Anyone who doesn't want to fill it out on a form voluntarily has that right - but it's basically a feel good gesture in America. We rank as the #1 Socialist State in the world, Free or not, like it, or not.
If anything, we should celebrate not giving it for a firearms transfer as one of the few things we don't have to. Don't get giddy over it tho, there's a catch - if the Federal Government issues the number, don't you think they can figure out who has it?
We are the ones giving it away freely in exchange for other things in life.