I wouldn't worry too much...
As others have said, before 1968, US law did not require serial numbers. Most guns had them, but a lot of the budget guns (mostly .22s and shotguns) did not.
Its not illegal to have a gun with no serial number. IT IS ILLEGAL to have one with the serial number altered, defaced of removed. This applies to all guns that had serial numbers put on them, no matter when made. Sadly, the way the law is written, it assumes that if you have such a gun in your possession, you were the one who removed the serial number in violation of law.
There have been people who have been prosecuted for having such a gun, when they weren't the ones who altered the serial number. If you do ever run across a gun with an altered, defaced or removed serial number, I'd contact a lawyer and have them make arrangements to surrender it to the authorities. (or follow whatever different advice the lawyer gives you)
I wouldn't worry too much about your old shotgun, dealers see them all the time, guns made before 68. I've had a couple myself. One was a bolt action .410, took it out of the stock, checked everywhere. The only marking on it was "2 1/2 & 3 in shells" on the barrel. Another one with no serial number was a Siamese Mauser bolt action. That one went to an FFL shop for some work, and the FFL said, "no problem". In the serial number box on the form, he just wrote "no ser number" and everything was all fine and legal.
If it never had one, you are ok if it doesn't have one. If it EVER had one, and now doesn't, get a lawyer, and be prepared to eat any investment you have in it, plus lawyer fees. Or risk jail. It's that simple.
Added: Its not illegal under Federal law. Check your state law carefully. There might be differences.