How should I have the caregiver pay for mom's groceries?

Depends if it is a private caregiver or an agency. Agencies have procedures for shopping.

NOT with a card. Have her pay and reimburse her. Or send her with cash and have her present the change and the receipt, and check purchases. Better yet, now switch to online purchasing for her yourself and pay yourself, getting reimbursed and keeping receipt and records.

You can preorder and prepay, have it delivered or caregiver pick it up.

You can give her a debit card preloaded with a budget or range of expenditure in mind.

You can leave her petty cash and a list and ask that the receipts be left in an envelope for balancing later.

I have done all of these with success.