This sounds like it should be super simple; and I can't believe I can't find documentation to support my assumption. I have a field, MDID (managed department ID) on the contact level, and I'd like that same value ported over to the same named field on the account level. I thought I'd be able to do something simple like a workflow rule and field update, but doesn't look like it. Should I be looking at the Process Builder or is there perhaps a simpler option.
Workflow rules can only affect the currently processed record. However, the Process Builder can indeed update a related record. When you create the Process Builder, start with "When a record is modified", choose "Contact" as the primary object, and choose to perform an immediate update on the related Account.
If Managed Department ID is the same for all contacts in an account, then you should use a cross-object formula field in Contact to reference the account's MD ID. That's the point of using a relational database; the child can inherit the parent's values.
To get a value into the Account MD ID field, you can make a conversion mapping of the Lead MD ID field to the Account MD ID field.