Possible? Yes. Absolutely, without a doubt. There are dozens, if not hundreds, of horse folk in the USA alone who'll sell you a dose of their prize stud's semen. Even more in the dog world, since there are proportionally more dogs out there than horses.
The first catch being *IF* you're someone with an established reputation as a breeder, and have a mare or bitch *THAT YOU CAN DOCUMENT*, and *THAT THEY APPROVE OF*. Joe Random off the street has exactly zero chance of getting anything other than a slightly more politely worded version of "Fuck off, shithead!"
Practical? Not really. As has been noted, you'll get a tiny bottle of stuff that's ready to be put in a mare or bitch - for horse semen, figure you'll get 5-10ml at max. Perhaps slightly more for a dog. You *WILL NOT* get the pint jar full of raw jizz that I have no doubt you think would be the case. It will have been processed - at absolute minimum, it'll have been centrifuged and the sperm portion (which is the tiniest fraction of *ANY* animal's ejaculation, no matter how much "raw material" they actually pump out) taken off and packaged, with the vast majority dumped down the lab sink as the waste material it's considered to be. It will almost certainly have had preservatives and stabilizers added, to the point where what you get will bear little resemblance to the actual stuff. By the time it gets done being processed, it'll be more egg-whites and additives than jizz, and no, you CANNOT ask for "the raw stuff" - Plain and simple, YOU WILL NOT GET IT. The very best result you can expect from trying will be getting laughed off as the total and complete idiot you'll have demonstrated yourself to be by the simple act of even asking.
And then there's the real killer: For horses, figure on a price tag *STARTING* at five thousand dollars, and easily going as high as half a million or more, depending on the stud. For dogs? <shrug> Dunno about prices there - I haven't had any experience with them the way I have with horses. Don't expect it to be cheap, though. I wouldn't be at all surprised to find out that a dose of semen from a popular stud would start at upwards of a grand, and finding out that some studs command prices in the tens of thousands would come as absolutely no kind of shock to me.
In other words, the chances of any random person getting a load of horse or dog jizz to use in satisfying your idea of a kinky sex session is somewhere on a scale that starts at "slim-to-none", passes through "Yeah, right - keep dreaming", and continues until it hits the wall at "not a fucking prayer - what the hell kind of stupid-weed have you been smoking???".