Federal Court stays immigration order

Well that's the tricky part. The branches all sort of rock paper scissor each other to keep everyone in check. I guess a judge could order CBP to not enforce the order, and if they do enforce it he could potentially hold someone very senior from that department in contempt of court. Additionally previous administrations would be afraid of impeachment by congress. Which is another remedy, but it seems unlikely with the republican majority in the house and senate.

But there is a twist, since the executive branch runs all the prisons, if the executive branch at the federal level is ignoring the rules it is unlikely that they will hold someone in custody for contempt.

And that is why people think it could become a constitutional crisis.

It's complicated. The President has broad authority in many matters; however the real brake on what he can do is the threat of impeachment, which here is zero. He has no control over the court system which would likely force others to follow their orders. If it were brought to the supreme court, it's hard to say what would happen, I suspect even with the split they would rule against him.