My mother has an evening sitter who I pay and manage. I created a daily log sheet for her to fill out and sign for communication.

kmich, if you SISTER is the POA, as I see you stated below, I am afraid there is little you can do because the aid is actually going to the person who has POA. That needs to be addressed at once. Simply re-tell the aid that Mom is not allowed to leave the home. Retell this sister that the Mom is not to leave the home. Get that POA fixed; continue to call one of many notaries; there are many who do inhome visits, and get that fixed. And on you go. Not much to do about what WAS done. Just restate and hope it doesn't happen again. You have an aid you like, Mom likes and that is a rare and lucky thing. I doubt there has been damage done.

I am so confused. So your Mom called for permission and got it. And then she went with the sitter shopping? What in the world did the SITTER do wrong? Mom got permission to go. Was Mom masked? We all are having to go out shopping. My partner is 80 and I am 78 and we have to shop at LEAST every few weeks. We wear masks. Your sitter is coming and going from your Mom's home. You do understand that exposes your Mom every single day? Because your sitter could have covid without symptoms. We live in Pandemic Times. We must do the best we can. But if you are going into Mom's home now with a sitter who comes and goes? You are exposed every single day.
Your sitter did absolutely nothing wrong. If you wish to make it clear to her that the Sister cannot give permission then please do so. Sit her down gently and say "I have to tell you that I am the only one who can give Mom permission to leave the home for ANY reason. Please be sure she doesn't try to do an end run round to my Sister. I so appreciate you; I know you will be diligent in this. There are many in the family at risk for covid and some suffering its after effects". I cannot imagine that a lovely chat with her, gently telling her this, would have any bad effect on her at all. In fact it would clarify for her what you expect because right now clearly she hasn't the slightest idea. She thought your Sister could OK this.

Who has POA for Health for Mum? If you do then the sitter has to accept the limitation of not driving your mother anywhere. But if you do not, then Mum even in early dementia has control over her body and can choose to go out.

Covid-19 further complicates everything.

Regardless of the POA situation you need to have a face to face or mask to mask conversation with the care provider and explain how serious it is that one she took Mum out against your expectations and two falsified the log book. Let her know this makes you question if you can trust the entries from other days and is also putting your family at risk of exposure.

Formally write her up and let her know her continued employment hangs in the balance.

Also offer her some tips and suggestions on how to deal with your mother when she is restless in the evenings.