I am looking for Nameplate extension similar to the one found here: https://www.soufeel.com/necklaces/design-your-name-necklace
Where can I get the extension or how would I add this functionality myself?
I can't say for sure whether or not this is an extension. It could just be custom programming that is specific to the site plus custom options/configurable attributes.
In any case, there are a variety of extensions that can be found with a simple google search for "Magento Personalized Products" or something similar.
This one seems to be the best fit for you that I found:
Here are a couple more that should work, but might be overkill for your needs:
Custom Product Builder - Buildateam -- This one is free but focused on images and only available for M2
Premium Online Product Designer - CMSmart - Does both M1/M2 (like Milople.com), but may go further than you are looking to go, making it more complicated than necessary, and it is a litle more expensive.
I would suggest a Personalized product solution for Magento 2. It provides the solution in many industries like gifts, crockery, furniture, curtains. As per your requirement, this will fulfill your all needs.