As someone who has done a lot of MT accompanying, I would say it's often not necessary, and sometimes positively undesirable, to double the vocal part, even if it's written in the piano part, as is often the case in collections of MT songs. If the singer knows the song, hearing their line on the piano won't be much help, and may be a hindrance if (as MT singers tend to do..) they vary the tune or rhythm. What matters most is to keep the bass and harmony going, and the rhythm of the accompaniment if that's important (e.g. "Tonight" from West Side Story has a syncopated pattern in the RH that complements the flowing vocal melody). You need to keep going in tempo at all times (except if there are rits, etc, naturally, in which case try to be guided by the singer) - don't stop or pause to correct mistakes.
Of course, it depends on the singer: if you're accompanying children or inexperienced adults they may need the help of the vocal part from the piano - as you get more experienced you will learn to recognise what sort of support a singer needs, and how to provide it.
Whether you have to play the whole song or just snippets depends on the length of the song, and the MD's (and others') preference, but I think you should be prepared to play all of it, just in case. Make sure you can deal with any tricky page turns and repeats. (There should be a special place in hell for people who bring illegible, unmanageable, badly-stapled copies to audutions.)
Good luck!