One element is tradition. The popsicle was supposedly invented by Frank Epperson when he left a drink mixture on the porch overnight in with a wooden stirrer in it. (Some historians have questioned this narrative, however, given that Epperson claimed this occurred in 1905 in San Francisco, but weather records show that it never got cold enough in the 1905 winter there for Epperson's story to hold up.) The choice of wood, according to this traditional story, was somewhat accidental, though given that plastics were not yet available in 1905, Epperson's main choices for stirring would probably have been metal or wood. (And obviously metal would be uncomfortable to hold in a frozen treat, regardless of whether this origin story is true or not.)
In any case, Epperson later patented his invention (including the wooden stick) in Oakland in 1923 and started selling them. Again, this would have been too early for plastic to be a reasonable alternative, so part of the reason for wooden popsicle sticks may just be that they are traditional and familiar to customers.
A business seeking to make a change away from such a tradition would need motivation, either an economic reason (a cheaper but still acceptable product) or an actual product improvement. Given that wooden popsicle sticks are intended to be disposable and already function well, economics would be a primary reason to switch, if there were a cheaper material. But Memj's answer is correct here. It's simple economics: wood is cheaper in this case, which is the reason it is used for various small uniform disposable items from toothpicks to chopsticks, utilizing technology that has been in wide use since the early 1900s. (For more information on the reason for manufacturing and process on producing small wooden items cheaply, I'd recommend Henry Petroski's book The Toothpick: Technology and Culture.)
It's also important to note that some businesses actually have experimented with disposable plastic sticks. The most common type were so-called Elsie Stix or Icetix, which were distributed by Borden Dairy in the U.S. They were intended to be collectible and manufactured to be used as toys after serving their purpose in the ice cream bar. (See photos and further description here and here, for example.)
I would note that these sticks easily solved potential problems brought up in some of the other answers: A series of holes allowed the popsicle to be frozen through the plastic and thus added to adhesion. And the holes also decreased thermal mass and heat transfer, which would negate any potential discomfort from holding a piece of cold plastic, which seems to be a strange concern to me anyhow.**
In any case, these plastic sticks were undoubtedly more expensive to manufacture, but they were marketed as a novelty item. And their interlocking nature and use as a toy required people to buy more popsicles to build with them, thereby providing an economic incentive to justify the added expense.
Without such an economic incentive for "standard" disposable popsicle sticks, and with wooden sticks being traditional, there's little reason for businesses to make a switch to plastic.
** More details about thermal properties: While some plastics are somewhat better at transferring heat than wood, there are many types of plastic which would not be noticeably uncomfortable compared to wood in a small stick, even if they were solid. Note that the thermal conductivity of many standard plastics is around 0.2-0.3 W/m-K, only roughly double that of wood, which is around 0.13 W/m-K. Compare that to most metals, which could actually freeze one's hand to them, which have conductivities which are hundreds of times that big. Specific heat is roughly the same for plastic and wood, and the density of most plastics is only a little higher than birch, which is a standard wood for popsicle sticks. Bottom line is that plastic popsicle sticks would absorb heat somewhat more quickly and stay colder a little longer, but not enough to produce significant discomfort for a small, thin piece of plastic.
But, theory aside, one can easily buy reusable plastic sticks for homemade popsicles -- just search Amazon for "plastic popsicle sticks". (Here's a review comparing 26 different varieties of plastic molds, almost all with plastic sticks/handles.) And I don't think I've ever seen a review complaining about discomfort in holding them. I've also seen multitudes of baby teething toys, for example, which are meant to be frozen and are made out of plastic: the idea of a "damp hand freezing" to a small thin piece of plastic seems quite unlikely.