I have done many wood trays with acrylic paint with great success.
I usually paint my tray first, either white, or some other light color as a background. Sometimes 2 coats are necessary for darker colors.
Then get a little makeup sponge - like the little wedge kind, something with nice dense foam. Then dip that into your acrylic paint in any color you want. (Do not dip the stamp in the paint directly or it will get too goopy.) Then lightly pat the sponge on your stamp. Apply very firm pressure when stamping onto the wood. Even though the tray looks flat, it is not totally flat, so you need to press firmly and evenly on it.
If you need to touch up something that did not come out you can use a little paintbrush.
Then just keep stamping away until your design is complete. My favorite way to write on them is with the metallic paint pens. It looks great and the finish is pretty with the paint. Remember to stamp something on the sides of the tray around the handle part so it looks "professional" and complete and tied-together from every angle you see it.
I usually seal it when I am done and it is completely dry. I use Anita's brand acrylic sealer. Any water soluable acrylic sealer will work fine. I brush it on with a foam brush, but I am sure you could get a spray. I get better control by brushing 2 or 3 light coats instead of spraying one goopy coat. It dries very quicky between coats.
Hope that helps. I have not done much with ink, except for one frame, but the ink really soaked into the wood and the design was very pale. I would follow others' directions on how they primed first to use with inks. But the paints work great!