A few points...
1) Appeal to whom? The NICs/FBI/ATF?
The only "appeal" process that I'm aware of is for persons that are
wrongfully denied purchasing a firearm (mistaken identity, identity theft, person with the same name as a listed felon, etc).
That process can take anywhere from 6-8 weeks (been there...done that).
If a person is continually denied anytime they attempt to purchase a firearm, then one option is to apply for the "Voluntary Appeal File" program...
http://www.fbi.gov/hq/cjisd/nics/vafbrocheng.pdf
2) At one time appealing directly to the ATF/FBI to restore ones 2nd amendment
rights was possible, however, the Clinton era Congress refused to fund the program.
3 It's not just felonies that can raise a red flag. Certain misdemeanors will result in a denial as well. If a misdemeanor conviction carries a punishment of 2 1/2 years or more (even though a lesser punishment is imposed), then it would result in a lifetime disqualification.
As an example, in 1994 MA increased the penalty to 2 1/2 years for a 1st offense DUI. It's very rare (if ever), a person receives that sentence (or
any jail time, but the conviction would still result in a denial.
Without knowing all the details, I think the OPs best chance would be to have the
charge expunged or if he's really gung-ho about it, apply for a pardon from the Governor (yes... people have succeeded in taking this route).
With that, I have a couple of questions for the OP.
Did you answer "yes" or "no" on the relevant part of the 4473 that asked about previous criminal convictions?
(if you answered "no", possible perjury charges could be filed against you. Wether or not you had forgotten about the offense or believed the charge/conviction was immaterial is irrelevant).
Did an ATF agent follow up and contact you regarding the denial (they visited the FFL and myself when I was first denied)?