There are many places to purchase wood veneer, but it is important to know which type of supplier provides which types of products and services before you make your buying decision.
Wood Veneer Specialists / Suppliers
In order to have any type of reasonable selection or service when buying wood veneer, it is crucial to shop at a wood veneer specialty company - one that sells only, or at least mainly, wood veneer. Rather than your choice of 2 or 4 species, wood veneer specialists sell hundreds of species, with many available in a variety of grain slicing patterns (cuts). They also offer a variety of backers for all of the species, as well as multiple sizes, and an array of grades.
Lumber Yard Veneer
The main reason to purchase veneer from a lumber yard is if you need it same day. Unless you live near a major veneer supplier, which is unlikely, this is generally the only way to get veneer in your hands immediately.
If you are able to wait a couple days, your options open up significantly. Lumber yards generally carry only a few of the most popular species, such as maple, oak, walnut, and cherry, and they do so only on one backer type. What you see is what you get with no ability to match existing projects or buy varying sizes, grades, backers, species, and grain slicing styles.
Big Box Hardware Stores Such as Lowes and Home Depot
The giant stores such as Lowes and Home Depot are often the first place people turn to when searching to buy wood veneer. Unfortunately, they generally leave empty handed. This is because these stores carry little if any wood veneer. A quick search of their comprehensive websites at the time of this writing reveals that Lowes has only two veneer sheet choices (both red oak in either 1'X4' or 2'X4' sizes) as well as three edgebanding choices (two are red oak and one is white birch). And Home Depot's website reveals that they sell zero veneer (only plywood), but do have a few choices of edgebanding, but still only white birch and red oak.