Nick here from Shopify.
There are already some apps and features in place which integrate Shopify to Amazon, Etsy, and eBay to Shopify stores. Shopify has some help documents on each of these which you can read below to see if they are indeed what your client is looking for with each one:
Should none of the above options be what you are looking to do, you can look into creating your own app as you mentioned. It's definitely possible to create an app for Shopify. There is a blog post from the content team going through all the steps in the process which you can read here.
Hopefully, the integrations you're looking for are above, but if not, it would be great if you could reply in the thread here with more details about what you need as it will be extremely useful for our development team to know and understand what store owners are looking for.
Looking forward to hearing from you.
All the best, Nick
No problem at all @AdamSmith. I wasn't 100% sure what your client was looking for but thought it was still worth mentioning the features just in case.
As well as the blog post I linked to you in the original reply there are some other guides and blogs which I feel you could find useful when building an app. There is a guide for building a Shopify app with Ruby and Sinatra which you can see here as well as a list of different blog posts here.
Hopefully, this helps!