Bullet weight: Proper deer hunting ammunition uses heavier bullets, generally over 100 grains. Although the .357 Sig has higher velocity than cartridges such as the .38 Special +P, .40 S&W, and even the 9x19mm Parabellum, it has comparable bullet weights. Heavier bullets will retain energy better upon contact, creating a more effective penetration.
Reliability: This is a common concern when using semi-automatic pistols for hunting. Semi-automatic pistols by design are prone to malfunctions. Even the most popular and reliable semi-automatics can and will malfunction and fail to eject the spent cartridge case. When hunting and the opportunity to take game presents itself, it's crucial to have reliable equipment so you don't loose your chance.
Lack of experience: Semi-automatic pistols are not an ideal choice for the novice hunter. In addition to concerns over reliability, a hunter is more susceptible to making a wounding shot or completely missing than a hunter using a rifle.
Many other cartridges out-perform the .357 Sig in energy, reliability, bullet selection and hunting reputation. If you're looking to use a pistol caliber for hunting, consider calibers like the 10mm, .357 Magnum, or even a .357 Maximum and pair them with proper bullet weight. Alternatively, consider a hunting rifle in a powerful caliber like .243 Winchester, .270 Winchester, .30-06 Springfield, and others in that class depending on the size of deer and environment you're planning on hunting in.