No

You don't have to look after that beyond the first few missions for the Minutemen (you don't even have to do these, if you don't want to). The first few missions for Preston and Sturges will have you setting up power, water and food in Sanctuary and from there onwards settlements can be largely ignored and depending on how you play the game. There are spoilers below. I'm not marking them up and you have been warned.

But...

You'll get Sanctuary early on and meet up with Preston at Concord and this will fully establish a settlement with people in it. This settlement is the primary base of the Minutemen and will remain as such unless you progress the Minutemen quest line and retake the Castle, at which point this becomes the primary Minutemen settlement but the original survivors will remain in Sanctuary.

In addition to these two settlements, if you continue with the Minutemen you will frequently find yourself in a position where Preston wants to expand by helping another settlement, which will then join your cause and potentially get attacked, prompting you for assistance (which you can ignore if you don't mind them dying or if that settlement is properly secured).

If you do not progress the Minutemen quest line and instead go with the Institute, you won't end up getting any additional settlements throughout the game.

If you follow the Railroad, they'll ask you to take a settlement as their new safehouse after one of their existing safehouses is destroyed by the Institute. This settlement doesn't need a broadcast tower as part of the mission and therefore won't generate a population of settlers that need looking after.

If you follow the Brotherhood of Steel, the airport ends up becoming one of your settlements during their quest line.

The additional settlement (or in the instance of the Minutemen, a settlement of your choosing) is required in order to build the Signal Interceptor which is used to infiltrate the Institute later in the game.

Can't prove a negative, but I've never seen any shipments from the vendors that I build. I've stocked several with tier 4 general store vendors and they still don't have any shipments of materials.

As for Bunker Hill, it becomes a possible settlement point after a certain quest:

The Battle for Bunker Hill

And there are shopkeepers there that sells some shipment items. It seems to be the only possible permanent way of purchasing shipments while on a settlement.

Update:

There are a few other possible settlements with a npc vendor that sells some shipments:

Abernathy farm and covenant. However for covenant, you need to:

Side with Dr Chambers in order to not have to kill the town.