Ringolong may have a reference I don't have. The way that the
regulation reads, it can be grounds but isn't automatic. What's also interesting is that they don't actually ask you about it. On the IACRA they ask about drug offenses but the instructions (if you dig deep enough) tell you not to include motor vehicle based alcohol offenses because you'll be asked about that on your medical. Of course, drone pilots don't complete applications for medicals.
It might be worth calling your local
Flight Standards District Office (FSDO) and asking them. Own your mistake and above all, be honest. The penalty for fraud is far worse than the DUI which, as ringolong said, is one year at worst. You can find your FSDO information
here.
It also can be grounds for the FAA to revoke your license once you have it. I know that doesn't apply to you, but just in case others find this later on. The regulation includes both issuing the license and later on. Like with the application, there doesn't seem to be a requirement to disclose it unless asked. Traditional pilots have to report it on their medical. Might be an FAA oversight so this may change in the future. Good luck!