My paintings are based off of the clear and close-up reference images that you provide. Below are some tips for capturing a beautiful photo of your pet with the idea of portraiture in mind.
The distinct colors and subtleties of your pet's fur are best captured under natural light. However, please be aware that direct sunlight may alter the appearance of some features, so it is best to wait for bright but overcast conditions. Also, try to avoid using flash, which can cause red-eye and color distortions. The exception to this is if your pet has a black coat, in which case bright sunlight or flash can help highlight the tones and textures that may otherwise be lost.
If possible, photograph your pet in the same pose you would like painted. Try going down to their level, as portraits usually look best from the side (three-quarter or frontal views) rather than from above. It helps to make your pet as comfortable as possible during this process, use treats and toys to catch their attention, or recruit a friend to hold them in place. If you have two animals it can be easier to photograph them separately. Because this is a painting, subjects can be combined and extra elements in the frame removed, as long as no important features are missing or covered up.
It is always good practice to take and send more than one reference image. Close up photos of eyes, unique markings, and accessories, are highly encouraged, as these really help distinguish your pet and lend personality to a painting. At the very least, please include an overall head-and-shoulders shot, taken from a reasonably close distance, that is not out of focus, color-filtered, backlit, or over exposed.
High-end digital cameras and professional photographers are by no means necessary for this process, most smartphone cameras are well equipped to capture the necessary shots. More than anything, remember to have fun, play around, and don't rush things.