You can search by title using the “title” param on the /admin/products.json API that Shopinet mentioned. As in:
/admin/products.json?title=,limit=250,page=1,fields=“id,title,etc”
I’ve found that some very simple matching is done. For example, searching with a title of “blue” will match a product named “blue shirt”. But I haven’t found a way to insert real wildcards… and the matching from the API isn’t as good as the matching demonstrated by Shopify’s own admin UI widgets.
So for now I’m using the “title” query param, but planning to pull everything into my own searchable datastore in the near future.