Is it possible to create a “ban hammer” tool that bans someone you hit with it?

Solution 1:

Firstly the term "Ban Hammer" is a meme (i.e. a joke), often depicted as an actual hammer. This is a representation of a "hard/fast" ban in response to certain behavior.

As for actually banning someone from a server, Bedrock is currently only using an Alpha Version of Java's Server Access functionality. Currently, you can only control permissions through a Whitelist:

To use the whitelist, it needs to be enabled in server.properties. When enabled, the server will only allow selected users to connect, and you will need to know the Xbox Live gamertag and (optionally) XUID of all players who want to join the server; players without an Xbox Live gamertag will not be able to connect. If your server is public and you do not enable the whitelist, you will not be able to ban players from the server.

A Whitelist is used to give access to people who are allowed on the server. If you want to ban them, you need to remove them from this list.

This answer provides a method that uses a taglist command to kick players that have been added to the list, which could be applied via "hitting" the player with an item.