Could you make a custom stamp with just 3D printed plastic? Sure! Plenty of designs out there.
TL;DR I just wanted to play with silicone and fuse it with printed plastic:)
I was playing with silicone caulk to make mini figurines using 3D printed molds and thought that the cured silicone was pretty flexible. So as part of my further adventure in 3D printing I decided to combine silicone with 3D printing again. Given the silicone is flexible I thought I would make a great substitute for rubber part of stamps and hoped that the flexibility would mean that the ink transfer would be better than a hard potentially uneven plastic. Using the plastic as a negative mold also means potentially finer details (/slight more flexibility) but the silicone may be too flimsy by then.
I don't have any stamp pads so the image you see is a result of roughly coloring it in with a permanent marker then stamping. Looks decent imo.
Rather than design a whole new handle I just took the one designed by Pumpkinwaffle and modified it to suit. The handle requires support but you could remove the column for alignment to print without support
Technical
Squeeze some silicone into the mold and use a thin object like a toothpick to spread it and make sure it is in all the little nooks without air trapped inside.
Add a bit more silicone (don't need much given it is quite thin) and pop the base on with the little nobs facing down.
Note: Some excess should come out the holes on top. If you have access to a centrifuge I find it helps really well to pack the silicone down. It also helps to spread the silicone fairly evenly to reduce bubbles trapped, but a small amount doesn't seem to affect it.
Leave to cure, demold and attach handle. I print with ABS so a little bit of acetone works great to glue them together.