re: Mulch for vegetable garden

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Wood uses nitrogen to break down. So, if you till it under, it’ll use the nitrogen in the soil. Sitting on the surface, it won’t rob nitrogen from your plants in the root zone.

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If you till any "brown" aka carbon material in, the process is going to use nitrogen. That's why you can't make compost with just brown material. You need "green" aka nitrogen. It's not the wood or brown material using the nitrogen; it's the bacteria facilitating the decomposition of the wood that is using the nitrogen.

Tillage causes an increase in microbial activity, which accelerates organic matter breakdown. That microbial activity is what uses up nitrogen.

If you don't till the wood mulch (or any mulch for that matter) into the soil, you don't have to worry about it taking up a significant amount of nitrogen. Tillage defeats the purpose of a mulch anyway.