How can I attach small wooden shelves to the inside of a wooden shadow-box style of frame?

Gorilla glue and screws from the back into shelves.

I’m guessing none of your knick-knacks weigh that much.
Like Tropical Willie said, Gorilla Glue should easily do the trick.

Another option is to make a 12×12 frame
that you could just pop into the box itself.

Riffing on @Nimis ‘s idea…

You could build a grid of shelves outside the frame—say 3 horizontal shelves connected at intervals by 3” long vertical partitions. No side pieces needed. If you glue that together outside the frame, then you could just pop it into the frame and no connection to the frame would be necessary.

Gluing end grain (what you get at the cut ends of boards) is kind of tricky because the glue gets sucked into the gain and there’s not enough left to make a good bond. It can help to give the ends a pre-coat of glue, let it sink in and dry, then make the final joint using fresh glue.

Drill pilot holes with a very fine bit, then attach thin screws. Wood glue works too, but don’t let it ooze out or it’ll mar the surface and show up when you go to stain it.

I agree with @HungryGuy. Gorilla glue doesn’t work as well as you may think. My husband used wood glue on anything wood and I can testify that you will need a hammer to get it apart once its dried. I had an entertainment center that was suppose to be only screwed together. When it was time to move, my friends cursed the day my husband bought his first wood glue. It needed to be taken apart to move. We couldn’t get it apart.even after we removed the nails.

I like @thorninmud‘s grid