You have to make the whole image in Illustrator.
You may find using Image Trace helpful.
.ai is just a "wrapper". What's in that wrapper can be vector or raster or a combination. If your print provider needs vector, you can't use the Photoshop file at all other than as a basis for possibly tracing it.
The best thing to do would be reconstruct things from scratch. However, not being familiar with Illustrator may present quite a learning curve.
But... it may be helpful to know why your print provider is insisting upon an AI file. For traditional commercial printing, Photoshop files should be fine unless they are just poorly set up. And often a print provider may ask for a different format just to make you go away. There are some legitimate reasons a printer may need a vector file though. So without understanding why there's an AI requirement it is difficult to be completely definitive.