Prevent rewriting multiple vars
This version tries not to change functionality. Added two functions that reduce code duplication, both methods make use of javascript's [](square bracket) property accessor syntax which allows the functions to call different functions based on the parameter passed in.

var addListener = function(selector, method, listener) {
    var selected = visualModules.activeModuleContainer.querySelector('.js-' + selector);
    selected[method + 'EventListener']('click', listener);
    return selected;
};
var showOrHide = function (method, direction) {
    var moduleOptions = visualModules.activeModuleContainer.querySelector('.module-options');

    if (moduleOptions) {
        moduleOptions.classList[method]('module-options--active');
    }

    removeBtn = addListener('remove', method, deleteModule);

    moveUp = addListener('index-up', method, move);

    moveDown = addListener('index-down', method, move);

    if (direction) {
        moveUp.direction = 'up';
        moveDown.direction = 'down';
    }
};
var toggleModuleOptions = {
    show: function() {
        showOrHide('add', true);
    },
    hide: function() {
        showOrHide('remove', false);
    }
};


This version could change some things but very minor.  This one always sets the direction properties even on the one that wasn't being set before to avoid needing conditional logic for setting those properties. It also does not assign values to variables from outside the scope of the code shown which was only left in above in case it was necessary.

var addListener = function(selector, method, listener, direction) {
    var selected = visualModules.activeModuleContainer.querySelector('.js-' + selector);
    selected[method + 'EventListener']('click', listener);
    selected.direction = direction;
};
var showOrHide = function (method) {
    var moduleOptions = visualModules.activeModuleContainer.querySelector('.module-options');

    if (moduleOptions) {
        moduleOptions.classList[method]('module-options--active');
    }

    addListener('remove', method, deleteModule);

    addListener('index-up', method, move, 'up');

    addListener('index-down', method, move, 'down');
};
var toggleModuleOptions = {
    show: function() {
        showOrHide('add');
    },
    hide: function() {
        showOrHide('remove');
    }
};