Is it correct to say "via a"?

Is it considered proper English to say something like this?

I called her via a telephone.

Or should the indefinite article be omitted entirely?

I called her via telephone.

If the indefinite article is to be omitted, are there any cases in which it should not be? Or, does it even matter?

If you look at the Merriam-Webster dictionary, two of the example usages for via are as follows:

He did some research via computer.

We went home via a shortcut.

So the second example shows you can certainly use a after via. ("We went home via shortcut" sounds wrong to me.)

However, I would say via telephone. This usage is closer to via computer, because telephone here refers not to a specific telephone, but to the general medium of telephony. Similarly, you would go somewhere via train or via superhighway, if you are talking about trains or superhighways in the abstract; but via the Orient express or via the Mass Pike, if you are talking about a specific train or superhighway.

If you are talking about a specific telephone, I wouldn't use via; I think you have to say something like "I called her on my cell phone."

Both “I called her via telephone” and “I called her via a telephone” are grammatical (“correct”) English, and they mean approximately the same thing.

The preposition via denotes “the way” something happened, either the manner in which it occurs (“we went via car”) or some entity that intervened in the action (“we went via route 89”). In these examples, the difference is that telephone without an article refers to the modality of telephony—that is, communicating using the telephone system, whereas a telephone refers to the specific object used to communicate. In this case, one implies the other—if you communicated using a telephone then you must have communicated using the telephone system, and if you communicated using the telephone system then you must have communicated using a specific telephone, so the phrases are equivalent.