OK - completions need a bit of help to get going - brew includes the definitions, but you have to prime the completions mechanism if you haven't enabled one yet.
brew install zsh-completions
autoload -Uz compinit
compinit
Press y when asked to ignore insecure directories
Then to test things out run the initialization:
autoload bashcompinit && bashcompinit
source /usr/local/bin/aws_zsh_completer.sh
I know - it says bash and you use zsh - you can type aws followed by a space and gla and tab and it should pause a bit and complete to the glacier command instead of file completion:
aws gla
Once that's set, you have to set up your .zshrc and likely fix the insecure compinit error by removing write permissions on some directories.