How do I find an agency that will take my aunt as a client and myself as her caregiver?

You first contact Medicaid to see if that type of program is available in your State. If so, I would think Medicaid has a list of participating care agencies. This program started in Pennsylvania a few years back. I see Agency Ads all the time saying they will train a person then hire them allowing the now employee to care for a LO. But understand, that LO has to fit Medicaid criteria.

I have mentioned this program before in my posts. I know PA does it because I get their TV stations. I think other States are following.
If a person qualifies for "in home" Medicaid, the State of PA allows a family member or friend to be trained by a care agency. That family member or friend, then becomes an employee of that agency. They are paid by the agency and receive benefits. They are then allowed to take on their relative as a client.

I’m not certain if this will be of any help in your area. You will have to check.

In our area. Easterseals assists children and adults with disabilities.

A friend of mine has a son who has disabilities.

Easterseals pays my friend’s mom to help care for her son. Maybe, your aunt can inquire if she can receive help from Easterseals.

You would be paid for your services through Easterseals.

easterseals.com

That won't happen, but you can have a lawyer draw up a legal contract where your aunt will pay you for a certain amount of hours each week, and where you'll have taxes taken out as she would be your employer and you her employee.

I suppose that you can do a search for agencies in your area and ask them if they will consider this type of arrangement.

Best of luck to you and your aunt.

Not certain what you mean.
Any agency will take your aunt as a client, but taking you as the caregiver I doubt can happen at all. I cannot, in fact even imagine what you mean.

If you work for an agency it would be unusual to assign you as caregiver to your own aunt. I suppose it COULD happen, but would be messy and unlikely.

I hope you can explain a bit more of what it is you are hoping for.