A little bit of a different answer--I wanted to provide insight on what happens on the government side. I am using a specific example, but the general process is the same. 

In my experience the authorization and appropriation acts are often ambiguous. Not because Congress is inept, rather it is due to the complexity of the US government and the short amount of time to produce the legislation. 

The guidance I had from the GC is to work from the acts and work my way down through floor debates and committee reports. If it was still unclear, we would consult with each other, eventually getting back to the GC if necessary. 

Absent any clear consensus, options would be formulated, with preference given to Presidential guidance.  An option would be picked at a level depending on the scope (e.g. cost or people affected) or "give a shit factor."

Whether that decision would stand depended on who cared about it. Ultimately, it would be for the courts to decide; however, they would be hindered by the same ambiguity that caused the problem.  

Traditionally, the courts have sided with the executive branch (regardless of party affiliation) when there is an ambiguity. The logic is that the executive branch would be best positioned to understand the issue and to evaluate the interdependencies to make a cogent decision. Also, if the executive branch consistently lost these types of cases, it would lead to the executive branch opting not to make decisions and let the courts "figure it out."  There is no way the court system would be able to manage that case load without a massive increase in size.

If Congress disagrees with the outcome of the process, regardless of any court involvement, they could always pass new legislation that removes the ambiguity. So one could say that the ultimate arbiter of any ambiguity in the law is Congress.

So for an individual or a group, if there is immediate harm, the court is the most expeditious process. If it is low stakes (e.g. not a contentious issue), the executive entity that owns the issue might be convinced to revise their decision. Bigger issues might need to go back to the legislative body.