An easy place to find several mammoth tusks is in the Halted Stream Camp.
You get a miscellaneous objective from Ahmren in Whiterun to retrieve his family sword, which may be located in the cave at the camp.
In the same area as the sword, there is a shelf with 5-6 Mammoth Tusks just laying there, ripe for the taking.
It's slightly North-Northwest from Whiterun's main gate.
The symbol on the map looks like two sticks forming an X.
gaming.stackexchange.com/users/14775/Niro
The probably easiest place to find the Mammoth Tusk is right outside Whiterun, at the Khajiit camp next to the stables or in Dawnstar.
Their caravan moves frequently, so you might have to wait for them to reappear, if they happen to be "on the move".
Ri'saad
the merchant who sits in front of his tent has one.
gaming.stackexchange.com/users/4182/DrFish
Yes, you can purchase them from some vendors, and find them laying about in other places.
gaming.stackexchange.com/users/4239/Doozer Blake
There is a tusk on the shelf behind the innkeeper of the whiterun inn, you can easily steal it to complete the mission in the whiterun marketplace.
gaming.stackexchange.com/users/10817/ruhsuzbaykus
There is a mammoth tusk in Phinis Gestor's quarters inside the Hall of Countenance in the College of Winterhold.
Wait until nighttime, when he's sleeping, and sneak in and take it.
gaming.stackexchange.com/users/15930/voxanimus
Also don't forget to find the Transmutation book in Halted Stream Camp if your going there, its a useful spell to have around for making easy money.
You don't always have to fight giants to kill a mammoth, sometimes you can find one separated from the pack in the wilds, just roam around a bit, or buy a tusk from a merchant, or find one yet again in the wilds.
gaming.stackexchange.com/users/14908/Lawful
If you want something tuned a bit more to exploration, there's a few dead mammoths in the swamps to the northwest of the continent, a bit off the beaten path to Riften.
I'm not sure if their spawn location is constant, but on the characters I've run through there
, I've always found at least one with a tusk.
gaming.stackexchange.com/users/1649/Artless
I found my tusk while roaming around in Dragonsreach.
Once you become a friend of the Jarl (basically, killing the dragon is enough)
you can go around taking some of his items.
I did that and voilà, made me not have to go through trying to kill a mammoth.
Like c'mon, freakin' unbeatable giants.
gaming.stackexchange.com/users/23086/xFaLLeN_Eternity
Go to Jorrvaskr (where the Companions are) in Whiterun.
Go in the front door and make a right.
Go up the steps and make another right before the staircase going down.
The railing preventing you from falling down the stairs leading to the lower level is on your left and there is a chest to your right.
Open the chest and there should be a mammoth tusk in there.
If you join the Companions you can take it for free.
gaming.stackexchange.com/users/18965/System Folder
In Tigre there's one in the meadary, in the far room upstairs, on a shelf.
gaming.stackexchange.com/users/23615/Ninja
I was able to buy one at Birna's Oddments in Winterhold.
I recall it was painfully expensive for a low-level character, but it was worth it to hear Isolde say "It's a fine day with you around".
gaming.stackexchange.com/users/137318/Tab Alleman
In Skyrim I have a quest to retrieve a mammoth tusk to help someone set up as a merchant. Of course, mammoths are generally being shepherded by giants. I do want to  kill a giant  at some stage but my character isn't quite ready for that yet. Is there any way to obtain a mammoth tusk without taking on a giant?
Can you obtain a mammoth tusk without fighting a giant?
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<the-elder-scrolls-5-skyrim>
gaming.stackexchange.com/questions/37940
gaming.stackexchange.com/users/1561/jwaddell
You can find several at Halted Stream Camp, at the Khajiit camp or in Phinis Gestor's Quarters. You can also purchase them from some vendors like Birna's Oddments.  It seems that someone can indeed find tusks by any other means, rather than fighting. The most common one is exploration.
People are mentioning different locations, where one can find tusks.