The IRS position is that, while a 1099 is only required to be issued by businesses, the IRS will not object if non-businesses issue a 1099-NEC since it helps with compliance. There are a few web sites that will allow you to file a 1099 and 1096 for a few dollars each.
That said, the housekeeper shouldn't need one. The housekeeper is required to report their business income and expenses from their own accurate records regardless of whether or not their customers give them tax paperwork. If the housekeeper lost track of how much you paid them, maybe they would be satisfied with a spreadsheet or a letter from you confirming the amount you paid.
