Can I get a permit for a project if all of my neighbors sign off on it, even though it would not meet city ordinance requirements?

The building permit process allows for a citizen to ask for a variance which may or may not be granted. If the building permit does not meet requirements per the review process the issue will be discuss with the property owner/s, at which time if they wish council will refer the issue to the Board of Zoning Appeals. The Board has a process which will be followed to determine if a variance will be given.