How do I find a caregiver to come to my aunt's house one day a week?

You're looking for homemaker/companion service. These workers do not do hands-on care. They clean, run errands, cook, take the client to the doctor or to social events. Medicare will pay for a few hours a week if her doctor says she needs it.
Or you could hire privately using a site like care.com and find one there. Advertise that it's homemaker/companion and no hands-on care. Set the hours.
As for the doctors appointments, explain to whoever her doctors are that she can only have appointments if they are made during her homemaker/companion's work hours. Most of the time they're very understanding and will try to accommodate her.
If you hire privately you can always see if the person will be flexible on the hours when needed. Agency help doesn't do this. The hours are set and that's it. Good luck even getting a fill-in if the regular agency caregiver calls out sick. Hire private. It won't cost that much. If she has to do driving pay a little more.

Its not going to be easy to find someone to come in when needed. Agencies usually require certain hours a day. But no harm looking into it.

If Aunt is low income, you may want to check into Medicaid in home services.

Contact an agency and tell them what you need and how many hours. IF they can accomadate that they will.
Sites might be useful for part time hours.
You could also check the neighborhood see if there is a mom or an older teen that would be interested. (drawback on this is the background check that should be done. And not sure I would trust a teen to drive a Senior to a Dr appointment)
Some areas have Senior Volunteer groups that would do this. They also often have volunteers that would drive to appointments as well. Check you local Senior Center and ask.

No, Medicare will not cover this type of "care".

This is a support forum for caregivers, not an employment/job listing site.