Right now, the answer is no. I believe you'll need to be a registered reader, and have your readers card approved for old manuscripts. You can see the getting a readers card section for more on how to do that. (Note that you'll likely need a suitable reason, study related, before they'd give you access). You can't just wander up and get access, sorry.
However, in the near future...
They're currently doing a huge amount of work on the "New Bodleian" library, part of which will see it renamed as the Weston Library. This work will include creating a permanent exhibition area for famous works. I'm not sure if any of the Magna Cartas will be in that, but they certainly get a mention on the hoardings outside the building works.
The work is set to finish in late 2014, for an official opening in early 2015. The Magna Carta has been around for many hundreds of years, so I'm sure it won't mind if you have to wait a little bit to come and see it... ;-)
For now, the Bodleian have changing small exhibitions in the Exhibition Room next to the Divinity Schools. I think it changes roughly every 4-6 months. They select an interesting topic, then put out (in cases) both old and new books on that topic, along with explanations and descriptions. Currently, it's on Great Medical Discoveries, but it only just opened so I haven't been to see it yet. These have always been good in the past though, so they're worth stopping in to see!