Generally speaking you can't do anything to stop them.
There are a number of things you can do to slow them down and most importantly to detect the tunneling so you can take other actions to stop them.
The most basic is detection, monitor the ground area around the vault for any seismic activity indicating digging or drilling. This is standard practice at borders to detect illegal tunneling and around vaults or other underground secure areas. There are a number of existing security solutions: Relevant Google Search. This is complicated if you are in a city or other environment that would have regular underground construction, so an isolated site would make this much more accurate, as well as making it harder to disguise the starting point for the tunneling.
In order to make the tunneling more difficult you can put the vault deeper, requiring more difficult tunneling to reach. At the extreme end, you could use drilling techniques developed from oil exploration and make a "vault" that is only a foot wide, but is several miles underground, and the contents can only be accessed via a long elevator like connection. This would limit the amount that can be stored, but it would make it really hard to get to.
In terms of making digging harder, you can site your vault in natural rock formations that would be harder to dig through (very hard rocks, multiple layers of different material, underground gas pockets, etc.) Or you could surround your vault with layers of complicating materials, like encircling the vault with layers of reinforced concrete, or pressurized water/oil bearing rock materials. These would make digging through them very difficult and are likely to allow detection when/if these outer layers are breached.