My wife and child are traveling from Sydney to Mexico City and connecting through Vancouver. We are both American Citizens with prior DUI convictions from several years back. Is it possible to be rejected in Vancouver, even though our final destination will be Mexico?
Usually, in the Commonwealth countries like Canada, Australia, NZ etc. if you're in transit and not exiting the airport, the immigration officials will not even look at you, so you shouldn't have any issues at all unless you want to leave the airport.
Of course you will also have to make sure your bags are checked in all the way to Mexico, because bag pickup happens outside the transit area.
EDIT: according to this, you don't even need a transit visa http://www.cic.gc.ca/english/visit/visas.asp?country=United%20States%20citizens%20and%20permanent%20residents so I doubt you'll have any issues.
Yes, it is possible to be rejected entry to Canada based on your DUI convictions. From Determine if you are inadmissible:
If you have been convicted of driving while impaired by alcohol or drugs, you will probably be found criminally inadmissible to Canada. But as of March 1, 2012, you may be able to get a temporary resident permit for one visit without paying the C$200 processing fee.
Now, whether or not you are required to enter Canada while in transit from Sydney to Mexico depends on a number of factors including which airline(s) you are flying on, and possibly even what time of day. See Connecting through YVR for more information.