What a fun quest. I love gardening and would be happy to share what I know about real estate. Most lots in Austin are 0.16 acres... a few hidden gems are a quarter to a third acre. A larger lot would allow room for both a garden, chicken coop and possibly a shed too. My house is in South Austin - 78745. We still have homes in this price range, but they are getting more difficult to find. However, if you're okay with cosmetic issues and a smaller house, we could find something in your budget.
In addition to Austin, you could consider outlining areas that have more inventory in your price point. The older sections of Round Rock, Pfluggerville, Kyle and Bastrop have great lots for big gardens and chickens.
Do you have a location preference? Such as, must be withing X miles of work, school, hobbies? This also helps me guide folks to neighborhoods to meet their lifestyles. The great thing about Austin is most neighborhoods allow chickens.
Best Wishes in Your Search,
I think this is a great question! Austin has a healthy culture of people that want to grow their own food and raise chickens. We actually have a chicken coop tour that occurs every year where the public is invited to see people's chicken coops!
To go back to your question, you are ideally looking for a home in an older, well-established neighborhood. You are also going to need to find a home without an HOA. Most HOAs tend to frown upon anything that would be considered as "unusual". Unfortunately, most of them consider raising chickens as part of that.
South Austin has some great choices where you can find larger lots and chicken-friendly neighbors. Oak Hill also has some good areas too. However, finding a home at this price point may be a challenge. There are homes available in that price point, in those areas, but they are most likely homes that will need some updating.
Good luck on your home search!