This might be an old question, but this answer may save other users time :
First you might want to convert the subtitle from WebVTT (.vtt) format to Subrip (SRT) format. The Subrip format is more widely supported by media players, especially KMPlayer (which doesn't support vtt format by default).
You can use Subtitle Edit (https://www.nikse.dk/SubtitleEdit or https://github.com/SubtitleEdit/subtitleedit/releases) to do the converting jobs. This software is completely free and open-source.
For your convenient, you might want to do batch converting a lot of subtitles. In order to do so, you might want to use Batch Convert.
Batch convert is accessible via Tools -> Batch convert...
Batch Converting process
You might also want to Save in source file folder so that you don't have to move the subtitle after converting.
After that, use KMPlayer to play the video. It should have subtitle now. If it don't, you may try adding the subtitles manually in KMPlayer by clicking Menu -> Subtitles -> Load Subtitles or just Alt + O.
If all of that doesn't work, you may want to checking out if the subtitle is disabled. To do so, click the Subtitle icon at lower right of KMPlayer (as shown in below image) :
Subtitle Icon
Make sure Enable is checked.
Enable menu