Getting a quality photo of your animal is key to getting a high quality portrait. I'm asking for clear, well light photo of your animal because getting your pets facial features and as well as their fur markings and patterning is key to an awesome portrait. Rather than a generic drawing of something that looks kinda like your animal. The magic just isn't there if the reference photo isn't clear.
Here's some pointers to getting a good image.
1- Have the lighting in front of the pet and not behind them. Take a photo at sunrise or sunset.
2- (pertaining to dogs and maybe some cats) Holding a treat in one hand and your camera/phone in the other ask them to sit (this is more for dogs) and with the treat in hand hold it in front of them slightly above head level, and snap your pic. The idea with the treat is that they'll look in the direction of the treat, a good photo is a dog or cat in a seated position looking slightly up, getting a snap shot of their ears, head and chest. Although that isn't the only good posture recommended.
3- (horses) Preferable with good lighting, usual in the morning or sunset. A relaxed horse can make for the best photos. Getting a photo just off center and as up close as possible including their ears, face and neck to show their off more of their features. Refer to examples here.
-Take a close up photo including their full face, ears and chest.
-Capture a good moment when they're laying down but looking attentive.
-Capture a photo at your animal's eye level or slightly above eye level.
-Best moment to capture a good shot of your pet is when they're relaxed!
*I hope these photos and tips here will serve as a few good examples and ideas! I definitely want to give you the highest quality portrait of your animal.