I have a page of writing that needs to be flipped for printing. ‘Flip image’ is there but greyed out.
If you want to “flip an image” you need to select the image to be flipped. But I don’t understand what you really want to achieve by stating “a page of writing that needs to be flipped” (you can’t flip a text page)
If flipping needs to be done only at print time, check your print driver. Some (e.g. MacOS ones) allow to flip horizontally, vertically or both. This can be used to print on transfer material (for custom T-shirts).
For more relevant help, edit your question to mention your OS name (and LO version).
If your printer driver doesn’t allow to flip the page when printing and you want to print just one page of reversed text you could File > Export select PNG, make a note of the page size, change resolution to 300 dpi, re–instate page size and click OK. You will now have a PNG image of your page which you can flip in a graphics editor (or Draw) and then print. It is a workaround for a one off page.