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I tried playing some MKV files I ripped on my Sony UBP-X700 and Sony BDP-S6500 players, but I can't get the subtitles to work. The subtitles work on VLC Player. Is there a way to play an MKV with subtitles on a Blu-Ray player?

The one thing you can try is to flag the "correct" subtitle track as default and forced. mkvtoolnix will allow you to do this through its header editor.

My memory is screwed up right now, but you have to flag the track you want displayed as both FORCED and DEFAULT to make sure all players will display it be default.

If I do that, then will the subs be forced, without a way of turning them off?

That will generally turn them on, yes. Assuming the playing app has the ability to turn them off, you can turn them off, but you are saying that it does not SEE them now.

Your complaint was that the player isn't seeing them now. Are you saying that the player SEES them when they do not have both flags, but will not play them?

It depends on the capability of your player. I don't have either of those Sony players so I can't speak to your specific issue.

The subtitles on a blu-ray are usually PGS subtitles. It is a picture based format. Not all players can display PGS subtitles from a .mkv container. A common format for subtitles is called SRT, which is a text based format. This format is often easier for players to display. There are various programs and services to convert PGS to SRT. There are also sites where you can download SRT subtitles for popular movies.

I wonder if either of your Sony's supports SRT subtitles?

I think dcoke22 is right with container-type support.
(UHD)BD players must support PGS because this format is used on those discs. Yours wouldn't be the only disc players that don't support PGS when in mkv container. Next to suggested srt that might work, you can also try to playback the ripped m2ts file instead. Chances are the subs will work (you'll lose chapter information though).