I did some Googling, and it seems that there's no way to do this with InDesign as of a couple years ago. But I'm hoping that a way to enable the recipients of my PDFs to insert their own JPEGs/PNGs/etc. has arisen, without dropping $300 on LiveCycle. Sorry if I'm omitting any important details, thanks!
If you can accept that your customer will learn how to edit PDFs, recommend to them some low cost PDF editor. Web search reveals many of them - at least, if RGB only is ok. CMYK capable ones are rare. Adobe Acrobat is the standard, but it costs a fortune. A low cost CMYK capable PDF editor is Serif PagePlus X9. It actually is a low cost rival for InDesign, but opens and saves also PDFs, both CMYK and RGB.
Warning: PDF editing easily deforms the layout to junk, if internal bindings get broken after attempting a radical tex reflow or object reordering.
Well, it really isn't an InDesign issue. It will depend on the options for editing available in the version of Acrobat your users own.
If you place an image onto a page in InDesign, and then export to PDF, when you open the PDF in Acrobat, the user should be able to select that image by clicking on it with their cursor. Then right click, and choose "Edit Image." That will access the Edit PDF tools. Right click on the image again, and there should be a menu option to "Replace Image."
I am running Acrobat Pro DC, so I can't test it with earlier versions of Acrobat, but I remember following a similar process to replace images using earlier versions of Acrobat Pro.