Search a content type and return JSON
I've seen many modules for this, but don't seem to see any that allow searching by field.


There are many modules which provide this feature, not necessarily out of the box, but with some customization.

I would say, the best module to start with this would be Restful.
I have personally used it for a similar use case for building the service endpoints which could be filtered using specific field values, and it work charms!
The documentation seems to have improved a lot since the last time I've seen it!

There are other modules present which help you create custom endpoint too. However, the ease of building such filterable endpoint is good with Restful as per my experience.