Mürren can't be reached by road — no exceptions. Here's how it actually works.
From Lauterbrunnen: cable car up to Grütschalp, then change to the mountain railway down to Mürren. About 20 minutes total.
From Lauterbrunnen: postbus (line 141) to Stechelberg, then the Schilthornbahn cable car via Gimmelwald to Mürren. About 30 minutes total.
Take the Berner Oberland-Bahn (BOB) from Interlaken Ost — this connects onward to both routes above.
Only in Lauterbrunnen or Stechelberg — there's no way to bring a car into Mürren itself.
You'll need to move it yourself. The Schilthornbahn provides luggage trolleys, but pushing one along the road is hard work in summer and nearly impossible in winter.
If you're staying at a hotel, almost every one offers an electric shuttle cart from the train/cable car station — ask when you book.