I have a construction site within zero airspace of an airport. Is there still the possibility to get an authorization to fly a very limited time lapse construction monitoring flight?

Yes.  If you can adequately address how you intend to insure operational safety, it can and does happen.  It will all depend on the quality of the waiver request you submit.  

The airport is covered by laanc. And when I try to fill out the waiver it comes back and says that, and I cannot proceed further. Is it at waiver or authorization I am looking for.

Call them.

It is a waiver.  You need to file it online.  
Sign in to your part 107 dashboard on dronezone.

The problem is it is covered by laanc.. And it won’t let me apply for the waiver, it defaults back to the top saying it’s covered by laanc. Which is correct but at zero altitude.

Trackerputnam Posted at 4-15 20:39
The problem is it is covered by laanc.. And it won’t let me apply for the waiver, it defaults back to the top saying it’s covered by laanc. Which is correct but at zero altitude.

Call them on Monday.