I did something similar for VHS SHHH tokens once.
I had access to recent fallen branch from some small bushlike plant. I wouldn’t call it a full tree, as it didn’t have good growth rings (which would have been nice).
I found it best to do laser engraving BEFORE any wood treatments (staining, varnish, etc)
I did also find that Etsy and Amazon have good supply. Search for “wood rounds”. Be careful of size. Measure out on a piece of paper the diameter you’re after. Images on these sites are VERY hard to visualize.
For engraving, I found the laser did an exceptional job, but lining up was hard. That was until I found a GREAT way to do it.
I created a series of circles of increasing size as an image on the laser cutter, like a bullseye. I then burned this into a thing piece of scrap that was clamped down.
I was then able to just use the markings to help centre the rounds on the bullseye, and use the laser software to only burn the pattern I wanted, centred on the same spot in software. It was great.
To get the best results, I painted the wood all in black, then sanded away everything except the recessed lettering with fine sandpaper, then varnished the wood.