So sorry for your loss.
I'm thinking that the 403b is similar to an IRA (it's equivalent to a 401k), and I think you can convert it to an inherited IRA (usually a 403b/401k can be converted to an IRA after leaving the job, but most people don't realize that). So the following is based off the idea that they are going to follow the same rules as an IRA. Obviously, contact Vanguard to discuss how to proceed with creating your BDA account and the transfer process (they are very experienced with handling these types of things).
The account must be transferred to you as an inherited 403b or IRA. Inherited accounts are usually labeled a Beneficiary Designated Account - BDA). If the account isn't properly set up and the distributions taken properly and in a timely manner, you could be forced to liquidate it in 5 years (and the money will count as income and taxed).
When the 403b was transferred to your dad, he received it as a spousal 403b, and wasn't subject to the same issues as a non-spouse (like you) will face. I know you'll not be subject to any withdrawal penalties so if you wanted to you could just take the whole thing out (and pay the taxes on it tho). You may have to take the required minimum distributions as your father was old enough to be required to take the RMDs, but definitely check with Vanguard to find out if it makes a difference if your mother wasn't at the age to be required to take the RMDs. Good news is even if you are required to take the distributions, they'll be really low as the amount in the account won't be crazy high, and they'll base the distributions on your life expectancy.
Vanguard will calculate the amount you need to take, and you can set the date to take the distribution, and also choose to have them withhold taxes out of it too. They'll send you tax forms you'll need too when the time comes.
Went through this a few years ago. Sister and I were beneficiaries, contacted Fido and sent death certificate, was walked through opening an inherited IRA (BDA, and usually says something about original depositor's name) and then sent a basic letter of instruction to adviser stating that death cert was sent, and according to the beneficiary designation on his account, I was a 50% heir, and to please initiate the transfer of my share from account number XXXX that belonged to my father to my BDA account (XXXX).
