Can I pull another wire through my existing conduit?

Is the fiber from another provider?  What kind of wire is in the conduit now?

Yes, the fiber will be from another provider. Right now it only has an analog landline phone wire. I would like to avoid additional trenching and know that the fiber and copper will not cause interference with each other.

I could use the phone vault or cut access into the conduit nearby.

It wouldn't be legal to break into the phone company vault, you'd be better off going into the conduit.  What I'd do is break the conduit, pull the existing wire out from the house location with a pull string tied to it.  Install a Y adapter on the conduit than pull your fiber and the existing wire back with the pull string this is hoping everything pulls easy and the conduit isn't rusted through if metal or damaged.  This way you haven't removed the phone company line, it will still be intact, as it is still phone company property.  The conduit more than likely isn't. 

We are in an interesting situation. I am in an HOA that owns the streets. The project is sponsored by the HOA. I finally got an answer from AT&T staff who told us that under those circumstances, they could not object to our re-using the vaults for other purposes, but would be responsible if it damaged the phone wires or caused other problems in the future.