Katy here from Shopify. Thanks for reaching out, and welcome to our community!
Great question! We have a couple of options to help you when creating an app for your Shopify store. This is a nice feature to add, as it offers your customers even more ways to interact with your brand.
One option involves using Shopify's open API to allow you to build your own app. You can do this by generating API credentials for your app from within your admin. We have a guide which walks through this in more detail here with the specific steps needed in this section here. This allows you to create a private app for use on your store only (i.e this will not be published to the Shopify app store).
We have a great blog article which you may find useful too: How to Build a Shopify App in One Week. Depending on your level / knowledge/ time available, it may require some more time, but the guide helps to break things down into sizeable chunks.
Alternatively, there are already some existing apps in our Shopify app store which allow you to complement your store with an app. This would save you from building an app yourself from scratch. These are third party applications, but are hosted with Shopify.
The route you take is indeed up to you, though do have a look and feel free to let me know if you have any questions.
Thanks so much for your reply, @Vadym!
I've chatted some more with our internal team on this for you. Apologies for any confusion here - when building a private app, you will not need to host on your local machine - though you will need to secure your own hosting for this.
There are however some options for this such as Heroku or AWS which you may like to consider to help you out. Heroku for example offers a free plan option.
We do also have a dedicated API forum board which you may like to chat in for specific help with this too.
I hope this helps you out with your build.