I think that's something you're going to have to talk to the humane society about. How shelters handle pulls to other shelters depends on their individual polices. Some of them will allow other certified shelters/rescues to take animals that are on the euth list for free, while other places will require the rescue to pay the animal's adoption fee before taking them.
If your rescue is not a certified nonprofit or anything like that yet, then I would guess that if they even allow you to take animals from them, that they'd require you to pay the fee. You'd probably have a better chance of being able to pull animals for free once your rescue is more official. But I'd worry about how you are going to pay to care for these animals if you can't afford adoption fees. Around here, at least, it's much harder for rescues who have not done the paperwork to become official rescues yet to collect donations.
If you want to help animals in your home but can't afford vet bills and fees, have you considered asking the humane society or another local shelter if they have a foster program? Usually with a foster program the animal(s) live in your home and you care for them until they are adopted, but the rescue they are associated with pays for the animals care - things like vet bills and food.
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