This is herat-breaking and I am sorry you have to go through this, seeing the birds in distress and agruing with your mom.
Agree with most folks here that I, too, want to save unfortuante birds but in this situation, the way you describe it, you would just be enabling more abuse since your mom would see your taking a few as a sign that she is doing right by the birds and she would get more anyway.
Before you call animal control, do you think it would make an impression on her if you explained that you can't take any of her birds because they are a risk to your farm because they are sick? It may be hard to disappoint your mom in her illusion that she has this under control but it may be an important first step.
As others here, I would also be very hesitant to introduce sick birds to my farm. They can have many problems that can contaminate your place long after they are dead - not to mention the heart-ache of watching some of them improve only to succumb to illness the next day. There is nothing like being up all night with a bird that improves and then it dies. It takes it our of you. There are mites and lice that can lay eggs and be almost impossible to eradicate once they are astablished at your place, on top of the diseases others have mentioned.
As others have also said, I would not wear the same shoes home that I wore there but keep them in a plastic bag in the car and change before getting in my car again. Shoes can carry diseases and parasites from place to place, as can clothes and tools.
I almost think you owe it to your family and future flocks at your place to protect your farm from being contaminated. That said, if you want to make the sacrifice to rescue, I am all for it, for the sake of the rescued animals, but not with the backdrop information that your mom will "just get more" that will suffer the same fate. Why not get the "more" straight to your place instead and have a healthy flock? Let your mom pay for your chicks as a gift to you and you pick them out from somewhere else, maybe.