City Council had a resignation occur which would make the city unable to meet the notice requirement of 45 days for the May election.

Under the Local Government Code 22.011, the commissioner's court shall order an election to fill the vacancy if petitioned by at least 26 taxpaying voters from the municipality.

My question is, can the commissioners court order a special election be held on a non-uniform day? Meaning, can the commissioners order an election be held on some other day that is not in May or November?

In rare situations, yes. But -- it must meet one of the limited exceptions set out in Subchapter A of Chapter 41, Texas Election Code. Examples: emergency ordered by governor, law outside Election Code requires election on another date, etc. -- See Ch. 41.