RAW, looks like not.
Arboreal Sapling is a standard animal companion that a Druid could normally select.
Mammoth Lord archetype says that the animal companion that you get has to be 'a megafauna appropriate to the Realm of the Mammoth Lords; see the What Counts as Megafauna? sidebar for guidance and examples.'
The sidebar does mention that the list is not exhaustive, but it doesn't include any plant creatures.
That said, I would almost certainly allow it. We are already talking about an Uncommon archetype from an AP - not something from the core rules. So I would expect GMs to use a bit of leeway to allow awesome things from their characters if they are going to allow Mammoth Lord in the first place. Allowing an Arboreal as a Mammoth Lord companion seems no less balanced than any of the other things that are on the list.
Leaf Order and then Order explorer with Animal Order ?
With Order Explorer, does the character obtain Animal companion ?
Yes. Order Explorer also gives you a level 1 feat that has the order as a prerequisite. Currently Animal Companion is the only one that meets that requirement. It is possible that in the future, there will be more than one feat that would qualify. But even then the player could choose Animal Companion because it does still meet the requirements.
Also, I would note that Arboreal Sapling is Uncommon and has Leaf Order as the 'Access' entry. That isn't the same as a 'requirements' entry. You don't have to have Leaf Order in order to get the Animal Companion. Having Leaf Order just makes it more of an expectation that the GM will allow the Uncommon option.
For example as Mammoth Lord comes from Quest of the frozen flame and Pine Leshy heritage also, I think that it's plausible for an arboreal companion as megafauna.
Yeah. So do I. But other GMs may not read it that way.
I'd just generally say no. Megafauna refers to particularly large animals, and despite the intellect and functionality of mobile plants, they are certainly not animals.
The Arboreal Sapling would, at MOST, be considered a Megaflora instead, and to be truthful, these creatures are far too small to fall into that category even if it were a thing for PF2 as only the most insanely large plants throughout history have ever qualified for that label.
If your GM thinks it is a cool enough idea that's well and good but by no means would this be allowed by RAW.