Can you apply wood with glue over stained areas?

my question is why don't people glue over stained areas? Will it not hold? I don't want to have to stain around all of the letters before or after gluing.

Staining a piece will reduce the ability of the glue to soak into the pores of the wood and form a good glue joint. I'll quote now from Titebond on using their wood glues:

Can surfaces that have been painted or stained be bonded using Titebond Wood Glues?

Most of our glues are designed to bond bare wood. Painting or staining a wood blocks the pores, keeping the glue from penetrating into the wood. The Titebond Polyurethane Glue may work for gluing together painted or stained surfaces, but it is necessary to remember that the overall bond will only be as strong as the bond between the paint and the wood. We recommend that all substrates be clean of any type of paint, stain, or sealer.

This would apply to any polyvinyl acetate (PVA) (i.e. yellow) glue. As noted by Titebond, polyurethane glue might hold onto stained or painted surfaces, but the bond will be tenuous at best.

What do you guys recommend?

Glue first, then stain. Or mask the areas to be glued, stain, then glue, and be sure to wipe up any squeeze-out.

There are a few Q&As here with related info on bonding wood with stain or finish on it, here's another Should I finish before, during, or after assembly?

Because of the specifics of your build here I don't see this as a duplicate however.

I don't want to have to stain around all of the letters before or after gluing. What do you guys recommend?

What I would do is stain the chest, and the letters separately if they are also to be stained, and finish both separately also (but don't get finish on the back of the letters if you can avoid it). Because of the glueing procedure I'm going to suggest you don't need to worry about the normal problems of glue not bonding to finished surfaces.

You're going to be glueing using epoxy. I would recommend you use one with at least a 10-minute setting time.

• Position each letter on the chest and lightly draw around with a soft coloured pencil or chalk.

• Inside these marked perimeters drill a few small shallow holes (1/16" or 1.5mm is sufficient). This will expose fresh wood and give physical texture for the epoxy to cling to.

• If you got finish on the back of the letters score them deeply with a sharp awl or a boxcutter, being careful to avoid marring their edges.

• Dribble epoxy into the drilled holes. Don't be neat here, you want a little epoxy sitting on the surface as well.

• Spread a little epoxy on the back of each letter and then position it, press down and give a little wiggle to seat it and squeeze out any excess glue.

• Now leave the letters alone for the epoxy to harden. You don't need to clamp them in place, if you do want to add some weight on top of each be very careful they don't slip out of position. Glue is very lubricative and given the slightest chance the letters will slip a bit I promise you.

Because of the finish applied to the wood already the epoxy squeeze-out will be very easy to wipe away and any slight traces left behind won't be visible as they would be if not removed thoroughly from the bare wood.