How to display output of command while also parsing it?

Sounds like a job for tee:

build | tee /some/file
ami_name=$(do_something_to /some/file)
deploy "$ami_name"

 deployArgs=`build | tee /dev/tty  | extractDeployArgs` &&
    deploy "$deployArgs" #won't run unless extractDeployArgs suceeded

tee /dev/tty will print directly to the terminal and pass the output to the next command in the pipeline at the same time.

(Feel free to replace it with some other file (or /dev/fd/"$someFileDescriptor" if you need want the side output to go to $someFileDescriptor))

In more advanced shells (ksh, bash, zsh, but not in dash) you can set -o pipefail to make sure the pipeline fails if any of its links fails (useful if extractDeployArgs can't tell from its input whether build suceeded or not).

tf=$(mktemp)
build | tee "$tf"
grep -Fq 'AMIs were created' "$tf" && ami=$(grep -o 'ami-[0-9]\+$' "$tf")

# you didn't say if `deploy` can handle multiple args or not.
# uncomment one of the following:
# deploy $ami
# for a in $ami ; do deploy "$a" ; done

If your interest is in a particular string of output you could run it like:

msg="$(build | grep -e "AMIs were created\|ami-[0-9]*")"

if [ -n "$(echo $msg | grep -e "AMIs were created")" ];then
    ami="$(echo "$msg" | grep -e "ami-[0-9]*" | cut -d ' ' -f 2)"
    deploy "$ami"
else
    exit 1
fi

The first grep selects only the lines in your output that report AMI's were created or ami-#. The output is checked for "AMIs were created", and if present, parses the ami-# out of your relevant output and uses it to deploy.