I'm playing Wold's Vaults, a 1.18 pack that includes both of these mods.

https://www.curseforge.com/minecraft/modpacks/wolds-vaults

When you hook a static chest up to an inventory connector it is only showing that the chest has 27 slots, so it will show 27 slots of the static chests inventory and then show nothing else. The inventory connector is only reporting that it has access to 27 slots as well and will not let you input items, despite there being many slots available.

After further testing, I don't believe this to be an issue with Tom's Storage, but rather Cloud Storage since similar issues happen with other mods as stated in the above linked issue.