You would not be applying for citizenship. Applying for citizenship requires that you have permanent residence (ie green card) in Canada and have lived here approximately 3-4 years. You should probably post this on another forum geared towards those interested in immigrating to Canada with a work visa, attaining permanent residence as a skilled worker or other method.
While I do not know what "the AWA issue" means, (Animal Welfare Act? Adam Walsh Act?) and any particulars are none of my business, if this has been a problem for your sponsorship of your wife in the US, where you are a citizen and have the rights and deferences afforded to citizens, I suspect it would be an even larger challenge in Canada, where you would be a temporary worker or permanent resident and ineligible to apply for citizenship for years.
Most people who have accepted criminal responsibility for offenses both serious and minor are deemed "criminally inadmissal". DUIs, bar fights, shoplifting, assault, etc. Even if it is not on your record, and perhaps even if you weren't charged, you will be asked if you have committed a crime. If the answer is yes, you will have many challenges getting across the border. Lying on immigration applications is not advisable and leaves you and your family vulnerable to expulsion. In order to apply for permanent residence in Canada, a US applicant needs to get fingerprinted and furnish a police certificate for every state where they have lived since they were 18. They also need to furnish an FBI certificate. It seems you have some record somewhere, or else it wouldn't be a problem for the USCIS.