Deb,
I've made that trip a number of times, although I was travelling from Monterosso. Unfortunately, if you want to minimize the travel times that will mean leaving VERY early.
In the same situation, I'd use a departure from Vernazza at 07:29, arriving Interlaken Ost at 14:28 (time 6H:59M, 3 changes at Levanto, Milano Centrale and Spiez). All of the trains after that have a MUCH longer travel time. When you arrive at Interlaken Ost, you'll transfer to the small Berner Oberland Bahn for the short (20 minute) trip to Lauterbrunnen. Before boarding the Berner Oberaland train, be sure to check the small display on the side to ensure that the car you're riding on goes to Lauterbrunnen (at times the train splits at Zweilütschinen). From there you'll use one of two routes to reach Mürren. If you need more information on the routes to Mürren, post another note.
You may enjoy watching this short video that shows the Berner Oberland Bahn......
https://www.youtube.com/watch?v=tWxRCZMZWrc