How to add a search to a non-shopify site

[Feije]

Hi all. I have a feeling that the answer to my question is "You can't do that" but I'm doing my due dilligence to make sure. 

My site, https://www.rusticosweaters.pe.ca/shop-online is created outside of any shopify themes. Instead, we have employed the "Buy Now" method/plugin and have generated lists of categories. In this way lists of products show on the site and can be added to a cart. The issue is now that my client has added a lot of products and is having a hard time pointing customers to a specific product on a given page considering they don't have unique URLs using this method. Is there any way to employ a search bar to narrow down the products by title?  Has anyone come up with a solution for this issue?

[OpenThinking]

I don't quite understand your setup. Anyway. 

I'd use the API and make AJAX calls to Shopify and display the products. Something like this

/admin/products.json?title=my_product_title

or the by using the Predictive Search API

/search/suggest.json?q=cream&resources[type]=product