Was the Persian army dependent on cornel wood?

It is unlikely that the Achaemenid Empire depended solely on cornel-wood.

That empire started out without direct control over those areas now known as natural distribution of that species. If ancient and current areas of distribution are even largely identical that means they build their empire just fine without cornel-wood or imported it. But take a closer look on the map and you see that natural distribution is not identical with human introduction of that tree elsewhere or indicative of some isolated populations.

Trading your enemy what he needs for his weapons is really an ancient custom, from Noric steel sold to the Romans, over to the USSR selling (unknowingly via proxies) the USA the titanium it needed for the SR-71 and everyone today selling or otherwise delivering weapons to the Islamic State.

The conquests of Alexander were quite quick. It is unlikely that the wood needed for weapons is only ever cut for use as weapons days before battle. If it is of strategic importance, they would have amassed a certain amount of stores of it.

If you do not have your preferred material at hand, you then use something else. The oldest example of wood used for such a purpose is according to Wikipedia spruce. The Romans considered cornel as the best material available and used ash most often for

A hasta was about 2 m (6 ft 7 in) in length, with a shaft generally made from ash, while the head was of iron.

So if you need wood and think cornel is best, you are better off using almost any other wood then to seize making replacement weapons. While cornel wood is fine type of wood, the original premise of this question overstates its military importance, while it may indeed be a small contributing factor over all.

While the Greek hoplites dory were made of either ash or cornel, it seems that only the extremely long Macedonian sarissa really calls for this type of wood.

Much more interesting in this regard is the effect an environmental crisis on the Greek side had at precisely this time:

History of crisis in wood economies: Classical Greece
One of the most famous crisis of a wood-based economy is what happened in Classical Greece, where trees began to disappear specially in the areas of Attica, Boeotia and Peloponnesus where indiscriminate cutting of trees for several uses, associated to drought and wildfires led to a severe lack of timber in order to build lances, shields, ships, etc. and to a slow but progressive weakening in military and naval power of the peninsular kingdoms in Greece, that were overwhelmed by Epirus and by the Kingdom of Macedon, much more fertile lands because of their rainy winters. This process arrived to the apex with the conquest of Greece by Phillip II of Macedon.

Whether cornel even has to be used for sarissas is also somewhat contested:

Lammert thought that the sarissa would have had a shaft of ash, as this wood was used universally by the Swiss and others in the great age of the pike in European warfare. Snodgrass supports this suggestion. The quality of ash which makes it so popular for spears is its combination of strength flexibility and lightness. Pliny (H.N. 16.84 [228]) tells us that “Ash is the most compliant wood in work of any kind, and is better than hazel for spears, lighter than cornel, and more pliable than service-tree (sorb); […] the elm would rival it were not its weight against it”
(From Nicholas Victor Sekunda: "The Sarissa", Acta Universitatis Lodziensis Folia Archaeologica 23, 2001.)

For a modern mathematical examination of wood types have a look at those characteristics which may have mattered more to the Persians, bow wood, and see that there are differences, but on a relatively smooth spectrum. The high specific gravity of cornel wood is in many ways a disadvantage. Using another type of wood would present only different trade-offs but might not be regarded as wholly inferior in every aspect.