First thing, you could turn the check if the checkbashisms tool is installed into a guard clause, shaving off one level of indentation:

if ! command -v checkbashisms >/dev/null 2>&1; then
    echo "‘checkbashisms’ is not available; it can be installed from the ‘devscripts’ package."
    exit 1
fi

The remaining logic can be simplified a lot with the xargs tool.

if git diff --cached --name-only | grep "\\.sh$" | xargs checkbashisms; then
    printf "(No Bashisms found)"
else
    exit 1
fi

I understand that the code you showed it only part of a bigger script and you still want to use the generated list of files. However, calling git diff repeatedly will have neglegible overhead and you can wrap the code in a shell function like this:

staged_scripts() {
    git diff --cached --name-only | grep "\\.sh$"
}

staged_scripts | xargs checkbashisms

Or even

check_staged_scripts() {
    git diff --cached --name-only | grep "\\.sh$" | xargs $1
}

check_staged_scripts "checkbashisms"

First thing, you could turn the check if the checkbashisms tool is installed into a guard clause, shaving off one level of indentation:

if ! command -v checkbashisms >/dev/null 2>&1; then
    echo "‘checkbashisms’ is not available; it can be installed from the ‘devscripts’ package."
    exit 1
fi

The remaining logic can be simplified a lot with the xargs tool.

if git diff --cached --name-only | grep "\\.sh$" | xargs checkbashisms; then
    printf "(No Bashisms found)"
else
    exit 1
fi

I understand that the code you showed it only part of a bigger script and you still want to use the generated list of files. However, calling git diff repeatedly will have negligible overhead and you can wrap the code in a shell function like this:

staged_scripts() {
    git diff --cached --name-only | grep "\\.sh$"
}

staged_scripts | xargs checkbashisms

Or even

check_staged_scripts() {
    git diff --cached --name-only | grep "\\.sh$" | xargs $1
}

check_staged_scripts "checkbashisms"