At least it makes sense: ammo would be hard to find some 200 years after it was last produced. Not like Nuka-Cola. Nuka-Cola is omnipresent. Ammo is more difficult to find in Fallout 4 because it can’t be crafted; it has to be purchased, and there aren’t many merchants at the beginning of the game.
Starting from Sanctuary Hills, follow the power lines south until you hit a small farmstead. Try not to kill everyone there. You can pick some gourds to plant later at Sanctuary Hills if you haven’t found any yet. There is a merchant here who will be able to sell you a small amount of ammo, but it’s not a lot, in addition to some miscellaneous objects. From there, you can go Southeast to…
The Drumlin Diner
Southeast of Abernathy Farms you’ll happen upon a quaint old diner that is beset by a couple of raiders. Fair warning: these raiders are fairly tough if you’re low level and playing on hard. You can attempt to intimidate them, but unless you have high charisma, it will end in a gunfight. Once you’ve dealt with them, however, you can go into the diner and find a merchant who carries an extensive supply of ammo. The Diner is the most comprehensive merchant you’ll find as a low level grunt.
It would initially seem as though you need to go to Diamond City before you find a merchant, but this would be a huge mistake, as Diamond City is halfway across the map and definitely not low level territory. Apart from this, killing tons and tons of raiders will generally get you an ever-increasing supply of ammo, but only for pipe pistols and similar weapons.