Your title said using API. It's not possible to get access to YouTube file URL from video ID by that. By mean, with file URL you can also download it, distribute content etc while it's against YouTube Terms. So there is no such API for that.
Since you're tagged with Javascript, there is possibility to get direct access to the file URL. This probably is not as you wanted exactly but it worked flawlessly from the developer tools console available in any browser. For details Here you go
const videoUrls = ytplayer.config.args.url_encoded_fmt_stream_map
.split(',')
.map(item => item
.split('&')
.reduce((prev, curr) => (curr = curr.split('='),
Object.assign(prev, {[curr[0]]: decodeURIComponent(curr[1])})
), {})
)
.reduce((prev, curr) => Object.assign(prev, {
[curr.quality + ':' + curr.type.split(';')[0]]: curr
}), {});
console.log(videoUrls);