5.8GHz omnidirectional antenna inside wood
Wood has a relative electric permittivity of 3 or 4 hence as soon as it closely surrounds the antenna, the whole electric field shape/magnitude would change and largely render the antenna useless at 5.8 GHz but make it more suitable for a lower operating frequency. There will also be significant losses in the wood but these are livable with but the change in permittivity won't be good.


  would coating the wood with lacquer worsen the situation considerably?


Google the permittivity of the lacquer and consider also that the thickness will be quite small so, compared to the effect of the wood, the added lacquer might make a small worsening of the situation.

Just in case follow up questions are raised based on what-if this or what-if that, the precise solution that you may want is probably only accessible to an antenna simulation software package so don't expect follow-up questions like 

"what if the wood was placed further from the antenna" 


to be answered.