Is it illegal to use Google's reverse engineered suggestions API in a new web browser?
Although I am no lawyer, here is an approach you can take ...
URLs are open game for discovery. The best invention comes from doing something new and different, I believe google agrees. If you have come across a URL that does what you want as a API query that you can take advantage of, do so, assuming you know what to do with that JSON output. APIs are fundamentally open unless the provider restricts it with appropriate authentication controls. You can embed a basic google search in many things without issue today via a simple Javascript query, or template HTML form submission. This would be no different in my point of view.
If you are doing this commercially, that is where you need caution. Google's search is generally open in the context that you may see a terms and conditions on the page when you visit https://www.google.com, however the agreement is fundamentally one way, and has limited enforcement to a general end consumer. Its more an admission that they are subjecting you to things they legally must disclose as you consume the open service.
https://policies.google.com/terms?fg=1
That being said, if you commercially use the open / discovered API and then increase awkward demand they may come after you for commercial damage - because to stop it, they likely will need to not only process thier API for the query string but other patterns in the HTTP post meta data. They do indicate that any interferrence with thier services directly (which this could be if commercial and significant) is something they take seriously.