Planar Shepherd
From Faiths of Eberron, is the only way I have found to Wild Shape into Outsiders. It limits you to Outsiders native to one specific plane. Certain planes have a good selection, notably Celestia (Celestials are very powerful for their HD total, and there are a lot of them in various splats).
If you just want to be a Druid who changes form into Outsiders, though
I.e. you aren't trying to use Divine Minion for infini-outsider, then there is a solution.
First of all, be a Half-Fiend with one level in the Savage Progression (or find some other Low-LA way to get the Outsider type, like being a Lesser Planetouched, or a Star Elf). Then, dip a level in Cleric, or qualify for the Contemplative (CDivine) prestige class. Choose the Alteration Domain(Dragonlance Campaign Setting). Take the Domain Spontaneity feat (CDivine). You may now exchange prepared Divine spells of 4th level or higher for Polymorph.
Congrats, you can be Outsiders.
Alternatively, wait until level 15 and be an Urban Druid. They get Polymorph Any Object.
A Psychoactive Skin of Proteus doesn't work as it explicitly bans outsiders.
Harder, and more cheesier alternatives include things like the Recaster (Races of Eberron), which lets you add a spell to your list of one level lower than the highest level of spell you know, and that spell can be from any class list. It requires 3rd level Arcane spells, and only advances arcane spellcasting though, so either Alternative Spell Source/Southern Magician or serious cheese would be required to get in.
The Spell Domain has the spell Greater Anyspell, but it specifies it can only be cast from the Domain Slot, not any other way - so it's essentially useless to you. Also only 5th level or worse spells, so only good if you're already an outsider.
Arcane Disciple could theoretically give you the Alteration Domain's Polymorph, but again it requires Arcane Magic and it changes those spells into Arcane Spells.
The Initiate feats might be able to help you, but they all specify 'add spells to your cleric list' so it's not exactly easy as a Druid to be having a cleric list. Rainbow Druidsnake? God knows.
Complete Champion has Domain Staffs, which are like a domain, in a staff. Cast spells from them like a runestaff, but wonkier.
The CO staple of a Drakehelm from the Explorer's Handbook (doesn't get more obscure than the Explorer's Handbook) straight up adds spells to your class list. It's a magic item. And completely ridiculous.
Complete Arcane has the draconic legacy feat which adds spells to your class list. Polymorph might be in there or arguable for one of the planar dragons or the other. You could get it through shenanigans. Probably. God, I don't know.
So yes, it's definitely possible. Easiest way is to just wait for Shapechange, the God-Spell, to come online at level 17. Or level 15 with Urban Druid and Polymorph Any Object. Before that you are going to have to pimp yourself to a god and go to Complete Divine, though, looks like.
Notably, Pathfinder has Planar Wild Shape, which basically shortcuts 99% of this effort by handing you the Outsider type on a silver planar-shaped platter.
They did away with polymorphing entirely though, in favour of set lists of abilities and enhancement bonuses to existing stats, so in the words of that famous guy, 'No dice!'.
No, by default. Your average Dungeon World dragon is an intelligent being like (or moreso) any human and doesn't count as an animal. The Druid is very much about nature and natural animals, not magical beasts or intelligent species.
But, that's by default. In Dungeon World, fiction rules all and in your group's Dungeon World you may have established that dragons are animals. Then it's fine and fitting that the Druid would know them as they know hawks and bears.
But if you haven't established that, say "no" to shapeshifting into a dragon—it would just cheapen the whole fictional underpinning of the move, and generating good fiction is the only reason the moves exist.
If, in the end, you do decide dragons are legit animals, I have one piece of advice: when the Druid misses on a shapeshift roll, make your move very hard. That's how the immense power of the move is balanced—riskier the form, the worse the consequences of a miss. If you pull your punches there, you'll end up with the Druid stealing everyone's spotlight and be a sad GM.
Best Answer
You missed one
This is false. Druids also have the move Studied Essence (page 105), which reads:
This is a bit vague. Perhaps each animal contains a bit of spirit which can be studied, Perhaps each animal has one or more spirit essences that must be studied instead. Ask questions and use the answers. Leafwillow is the expert here; ask her to find out.
Spirits
The first thing you have to decide is what the nature of a blink dog is.
Blink Dogs are, according to page 265, unnatural creations of a sorcerer lord that got out and started breeding with the local wolves and wild dogs. Based on your description, it seems you've amended this to be a bit more fey-magic based. Great! Embrace the fantastic, then Play to find out what happens. This means they're likely animals with something altered and unnatural about their spirits.
Druids are all about spirit magic.
So when Leafwillow comes across a wounded Blink Dog and helps her party tend to it, she is in a pretty unique position. She know more about tending animals than the rest of her party. She is also uniquely aware of the spiritual damage it may have sustained from the equally spiritual Displacer Beasts. During its brief time in her care, surely she would have taken the time to tend the whole creature, not just its body, and thus spent time in contemplation of its animal spirit.
In summary, this lets Leafwillow become a Blink Dog because she's had time to interact with and study one. It develops your world more as the group learns about how spirits work. Finally, it provides a limit you seem to be looking for as a DM on why she can't just turn into all kinds of fantastic creatures.