I did a preliminary search and these rentals either accept pets or say "Subject to restrictions", which means most likely yes but the owner wants to know specific information before allowing the pet. If you want more specific help, or have more questions, you're welcome to contact me.
Keep in mind the rental market is ridiculous right now - prices are a lot higher than a mortgage payment and often rent in under a week. You may need to decide you will rent a place sight unseen for now. You can apply online for most rentals. I can also tour and send a video of a place - but only after the pets are approved via email. Remember, wherever you end up it is just temporary, for a year. You can always move again next yeat and be able to spend more time shopping in person. If you have this mindset it makes it a lot easier to decide on a place and have something in contract before you arrive.
Take a look at these rentals - this is the best you can do right now for distance and price. The pets are heart-wrenching, I know. I considered a move to West LA and did a lot of research online and was able to find a place that didn't care how many pets i had as long as I paid all the money (and it was a lot). But it definitely was limiting - you are blocked from about 80% of available rentals by property owners who have been burned in the past by irresponsible pet owners. A few bad apples ruined it for the rest of us.
I also think that you should consider a house. Typically in your type of situation the best deals are found in ones own network of people they know. In your case this is likely unlikely which is why a property management company would be the way to go. You'd probably need to count on making a pretty hefty deposit due to all the pets as well though.
If you want to message me privately I can give you some great property management contacts around here.
Wish you the best!