Is the term "walking across a tarmac" grammatically correct?

Oxford Dictionaries Online specifically references the tarmac

(the tarmac) A runway or other area surfaced with tarmac.

Similarly Collins

a runway at an airport: on the tarmac at Nairobi airport

The examples given are for discrete items or examples of a particular tarmac, not a reference to an undefined mass of material, such as pass me the sugar

It appears that tarmac can be a countable noun, and as such you should be able to use a tarmac. Consider the plural usage.

On tarmacs round the world, new technology is improving their durability.

This is different from the usage of the definite article with an uncountable noun. When referring to activities at several airports on the ground cannot be made plural (grounds means something different); on the tarmacs can be, although the singular mass usage could also work.

In the example you quoted, "a tarmac" is fine. Whether it should be "tarmac" instead of "a tarmac" depends on the intended meaning, specifically whether the author was referring to the type of material ("walked across tarmac") or the type of location ("an airport tarmac").

Normally you would use "the tarmac" when referring to a specific tarmac, however in this case it is using an ambiguous reference, similar to "an airport tarmac" or "any airport's tarmac." It seems no different to saying "a road" to me. Tarmac is definitely countable when referring to a type of location.