When a statute gives the president the sole authority and discretion to declare an emergency, then what is the "illegitimate" means you are talking about?

how about he declares an emergency, Congress passes the joint resolution to declare said emergency ended, Trump vetoes it, the veto is over-ridden by a supermajority in both houses, and then Trump keeps ordering executive branch people to behave as if the emergency is still in effect.

A different example would be if Trump goes on TV and says he's declaring an emergency, but doesn't ever actually do the paperwork, yet gives orders as if he had done so.

My question presumes that there isn't lawful authority to spend the money in the manner it is being spent. I understand there are all sorts of ways that the executive branch *could* have authority to spend different pots of money, but I'm not asking about that.

What I want to know is what happens when an instruction to spend money which is clearly over the line is given by the POTUS to the SecDef, and from the SecDef to the next underling, all the way down the line to the contractor doing the work, and everybody goes along. Which links in the chain are in legal jeopardy and what sort of jeopardy are they in? Is the project manager who approves the contract for the work the only person who actually mis-used federal funds, or did his/her boss who instructed them to award the contract also do so? How about that person's boss? How about SecDef? What about the contractor who bid on, and was paid to complete, a project without a legitimate appropriation?

The oldest current emergency declaration dates back to Jimmy Carter and keeps getting renewed.

You may find many of your answers here:

https://www.lawfareblog.com/emergencies-without-end-primer-federal-states-emergency

"Emergencies Without End: A Primer on Federal States of Emergency"

If needed, yes, Congress can obtain injunctions against specific officials who would be in contempt if they violated an injunction.