This mod adds tabs to the player inventory for nearby chests and other storages.
Mod spotlight

If I understand correctly to add compatibility You would need to "Chest tabs are tabs belonging to chests that are able to double up along the horizontal axis. To register your chest, pass the block reference to TabProviderRegistry#registerChest. Chests that do not match vanilla chest behavior should not use this method." - https://github.com/cakewhip/inventorytabs#chest-tabs

BYG doesn't add Chests?

Not sure how this is relevant to BYG as we do not have chests of any kind

Yes, sorry for wasting Your time. The chests I was talking about come from BetterNether. I thought I deleted BetterNether because it isn't compatible with jellysquid3's Hydrogen mod but instead I removed Hydrogen. I should have checked twice.
Can I ask do You plan to add chests from your wood types?

At this time, we do not