Hi Vince....and welcome to TFF!
Are there any other markings on the gun? Particularly, any proof marks? These are often hidden under the forestock.
I haven't been able to find any information on a "Finch Arms Co.".....but that's not unusual. From the 1880's until the 1930's there were many, many thousands of inexpensive shotguns sold under many hundreds of "private label" trade names.
These are generally known as "Hardware Store Guns" or "Mail-order House Guns".
Many were inexpensive Belgian imports.....they'll have proofmarks, usually a Crown over an ELG.
But many were also made here in the U.S. by such companies as Crescent, Stevens, and Harrington & Richardson.
Crescent alone is known to have made shotguns with over 120 different private label trade names.
It's possible that there was a small company called the Finch Arms Company that made shotguns, but this is more likely a private label mail order or hardware store gun.
Depending on when your gun was made, it may, or may not be safe to shoot. Many of the earlier guns were designed for blackpowder and are not safe with modern loads.
If you're thinking of using it, by all means PLEASE take it to a competent gunsmith and have it checked out to see if it's safe to shoot.