How big if the piece of wood? Can you see it? How far down is it?

They have grabber tools at Lowes, Home Depot, and Amazon that might work.

Use a mirror and light to see if your retrieval attempts pushed wood into vertical line.

Yup, Grabber tool.

Turn off water supply to toilet. Flush one or more times so the bowl is as empty as possible. Get out your vacuum (preferably a wet-or-dry shop vac), remove any tool from the end of the hose, turn on vacuum, push hose end down into drain. See if the wood comes out. This worked for me when someone tried to flush a popsicle stick down my toilet and it got stuck crosswise in the drain.

I would not call a plumber.  They will charge at least $750 for something you can do yourself.  Save $$$$ by DIY.

Sounds like the toilet is off already. Shopvac should have enough suction to grab a hold of that wood.

If not a hanger with a bent end and a mirror should be able to hook that piece of wood and pull it closer to the opening so it can be grabbed.

I wouldn't use a grabber it will push the wood further, not to mention the flexible grabber doesn't have too big of an end-grab.

Have fun fishing!

Nonsense!

One Example

On Submarines, we called these "Bilge Grabbers - retrieved many an object dropped in the bilges!