The FCC has options for inspection. The 4th Amendment insures your rights to be secure in your person, property and home. The FCC has to get a search warrant if you revoke your consent. There are particular caveats to this situation, as Doc mentioned. But it is different for CBers and Hams.
CBers don't run the risk of losing their license - of course - since it isn't licensable. Just fines. Can they confiscate illegal CB equipment? Well, only if you let them in. The 4th Amendment is not only against illegal searches, but illegal seizures as well. They must have a warrant to take the stuff, or tell them to take a hike if they fail to produce one before they attempt to enter.
Now, as previously indicated, a Ham can lose his license for not allowing FCC inspection of the station. But, you can protect your 4th amendment rights by refusing access.. This was the point the FCC offical failed to make before she was fired. As a side note, she may have grounds to sue for infringement of her First Amendment right to freedom of speech for being fired because she addressed stood up for people's 4th Amendment rights. I'll have to ask my bro about that one...
The FCC doesn't need to go into your home and examine your gear either. To allow them do this, is to allow them all of the additional evidence in a courtroom to sink your boat. You have the right against self-incrimination (5th Amend.) which is implicated when you make any admissions (verbal or otherwise) regarding the legal status of your equipment. They already have the info they need from their equipment in their van - which tells them (exactly) where you are, how much power you are using, how long the transmissions are, what frequencies you've been operating on, and so on. They have everything. Do NOT think for a moment that you can send them packing empty-handed. They already have what they need with the data. You are only kidding yourself.
The thing is, to see your equipment is just a verification of the data that they have already gathered.
But - it amounts to an admission of guilt by refusal to comply - which your Bill of Rights have provided for you. Don't relinquish it -losing your license may be the cost - but you are standing for your rights. You may very well lose your license either way - in much the same way that one can automatically lose their drivers license by refusal of a blood draw or breathalyze for a drunk driving offense. They may not win the conviction for the charge; but they have been punitive by removing the privilege.
Personally, I think that woman said and did the right thing. She hung on to the essential meaning of your 4th Amendment rights by declaring what should be already obvious to us all. She reaffirmed it - though it cost her - her job/position. Furthermore, is the 'privilege' of radio use just that? By taking it away - does it impinge on your right for free speech? No one that I've read has suggested this, but it can be so. Many years ago, the only way people could discuss politics and their gov't was in public houses - or 'bars'. The numbers of bars has greatly diminished, so that there are fewer avenues for people to speak about gov't - their gov't!. As far as my opinion goes, we have that right and it shall not be abridged or infringed.
Especially abridged!