You'll first want to enable RFQ so that your customers can see this option on the storefront. You can do that in the admin by going to Config > Order Settings > Order Settings.

At the bottom of the page (see above) you'll see the check box to enable it. Also be sure to select the correct email template.

Save your changes.

Now with RFQ enabled, the "Request Quote" button will appear on the cart page, as seen above.

Add a few items to the cart and proceed to the checkout page and select the Request Quote button to send the admin an email.

You should receive an email from your store, and in that email should be a link to the order: View Order Details in Admin.

This will then take you to the user's shopping cart where you will make the necessary changes to the order. Be sure to hit the Update Prices/Quantities button at the bottom of the page which will save your changes.

NOTE: The system does not automatically notify the customer about their updated cart. (See Below)

Next, you will want to send out an email to your customer showcasing the adjustments you made to the order. The email template you use here will be one that you create on your own.

Alternatively, you can just obtain the customer's shopping Cart ID from the shopping carts page: Orders and Shipping > Shopping Carts in the Admin like shown in the illustration below and provide your customer with that info.

The user will then visit the site, go to their Shopping Carts page under the My Account section and load the cart that corresponds to the Cart ID that you provided them with to complete the order. (See below).