The comma is exactly what you use to avoid ambiguity.

In the case of your first example, the absence of a comma before which marks the which-clause as restrictive—that is, the sentence refers only to those [files from the inquiry] containing embarrassing material. It is called restrictive because the sentence's reference is restricted or limited to those files identified in the which-clause. Only the files containing embarassing material were destroyed; other files were left untouched.

If you put the comma in, as in your A. example, the which-clause becomes non-restrictive—that is, the sentence refers to all [files from the inquiry], and the which-clause is just an added description of those files. All the files were destroyed, and they all happened to contain embarassing material.

With a non-restrictive clause you cannot use that—you must use which.

With a restrictive clause you may use either that or which. Some style guides (and some editors) claim that a restrictive clause should not use which but only that; but this ‘rule’ is not universally observed even in the most formal writing. It is only a ‘recommendation’, which you may follow or not, as you please.

Sentence A means the department destroyed all the files from the inquiry, and those files contained embarrassing material.
Sentence B means the department destroyed only those files found during the investigation that contained embarrassing material; other files found during the investigation were not destroyed.

In sentence A, which introduces a nonrestrictive clause, while in sentence B, that introduces a restrictive clause.

The NOAD has a note about nonrestrictive/restrictive clauses.

In writing, a restrictive relative clause is not set off by commas, and that is the preferred subject or object of the clause, although many writers use which and who or whom for such clauses.

This reminds me of a comic in my elementary school planner. It looked like the following:

Dog, eating chicken. [picture of a dog, sitting at a table and neatly eating a chicken breast]

Dog-eating chicken! [picture of a giant drooling chicken chasing a dog and trying to eat it]

So obviously, though exaggerated, the intention is to show that punctuation can have a huge impact on the sentence. :) In your example:

The department shredded all the files from the inquiry which contained embarrassing material.

The department shredded files. Which files? All the files that were 1) from the inquiry and 2) contained embarrassing materials. This is exactly the same as your modification B, which switches "which" for "that"--they are interchangeable here, and mean the same thing.

Now your example A means something entirely different:

A. The department shredded all the files from the inquiry, which contained embarrassing material.

Here the clause after the comma, "which contained embarrassing material", is simply extra information added to the sentence. It would mean the exact same thing if it only said "The department shredded all the files from the inquiry." The clause after the comma only exists to add extra information; the sentence stands on its own without it. This sentence means that all files were shredded, and all files contained embarrassing material.

The comma is used in writing to convey the idea that the which clause does not go with the immediately preceding element but with something more distant. However, this does not eliminate the abmiguity from the spoken language. There isn't any way to pronounce the comma to eliminate the ambiguity.

Semantics of the sentence eliminates the ambiguity. It is documents which contain embarrassing material. Inquiries do not contain; they are carried out and produce findings.

Language is inherently ambiguous. For instance:

John ate the sandwich in the fridge

Did he eat it in the fridge, or did he find it in the fridge? Both parses of the sentence are valid syntax, but only one makes sense.

One way you avoid ambiguity is to make the meaning so clear that there is a single correct interpretation, and the others are ridiculous.

Another way you can avoid ambiguity is to basically embrace it: there are two interpretations, but it basically doesn't matter which one you choose.

So for instance, your original sentence is works because inquiries don't contain. Suppose we have another noun there instead of inquiry:

The department shredded all the files from the box which contained embarrassing material.

Is it the files which contained embarassing material, and the box had other files which did not? Or did the box contain nothing but embarassing material?

Unless this subtle distinction is somehow very important in the overall message, who cares!

If it becomes important, the people who are communicating can seek clarification. Were there other files in the box, and what happened to those?

The department shredded all the files from the inquiry which contained embarrassing material.

This sentence should probably be restructured for clarity. I believe the intention is the FILES contained embarrassing material, but the sentence reads as if the INQUIRY contained embarrassing material.

I would restructure to something like:

Due to the inquiry, the department shredded all files containing embarrassing material.

If the inquiry contained the shredded files:

The department shredded all files containing embarrassing material submitted with the inquiry.

Or

From the inquiry, the department shredded all files containing embarrassing material.