Is it possible to create your own ammo?

The only special ammo in fallout 4 you can create is for the syringer rifle.

However, the shops and the wandering caravans tend to hold a lot of ammo.

Random caravans will show up in your settlements, such as sanctuary. Also the weapon dealers in Diamond City and Good neighbor tend to carry a lot of ammo.

Finally there is the scrounger perk that lets you find more ammo as loot.

It's worth mentioning there's a mod for this: Craftable ammo.

Mods are currently only supported for PC, but will be supported for consoles(!!) in early 2016.

You can craft ammo with the Contraptions Workshop DLC using the Ammunition Plant.

I have yet to try this in-game myself, so this answer isn't very detailed, but I'm surprised no one has written a nice answer about it yet, so let me at least note it.