Using moment.js with the moment-timezone add-on, you can do the following. The results will be consistent regardless of browser or operating system, because the abbreviations are in the data provided by the library.

The example results are from a computer set to the US Pacific time zone:

var zone = moment.tz.guess();            // America/Los_Angeles

var abbr = moment.tz(zone).format(z);  // either PST or PDT, depending on when run

DISCLAIMER

Time zone abbreviations are not reliable or consistent. There are many different lists of them, and there are many ambiguities, such as CST being for Central Standard Time, Cuba Standard Time and China Standard Time. In general, you should avoid using them, and you should never parse them.

Also, the abbreviations in moment-timezone come from the IANA TZ Database. In recent editions of this data, many invented abbreviations have been replaced with numerical offsets instead. For example, if your user's time zone is Asia/Vladivostok, instead of getting VLAT like you may have in previous versions, you will instead get +10. This is because the abbreviation VLAT was originally invented by the tz database to fill the data, and is not in actual use by persons in the area. Keep in mind that abbreviations are always in English also - even in areas where other language abbreviations might be in actual use.