I was wrong in the earlier version of the answer and 1.1 version of the API does support this via GET request.
The basic request would be: https://api.wordpress.org/plugins/info/1.1/?action=query_plugins&request[browse]=popular
And you can add more parameters by sticking with request passed as “array” (in GET interpretation of).
See the [poor] documentation in Codex and links from there for more detail.
For in–PHP way, independent of WP core I made a WPorg Client library which implements this, among other things.