I'm making a small refrigerator with a Peltier module but I can't find a heatsink. Will it work if I only use thermal paste and don't use a heatsink?
The Peltier module works by creating a thermal gradient when current is pumped into it. One side will get hotter than the other. The Peltier's heatsink removes hot-side heat to the air.
In this, the Peltier is just like an air conditioning unit with the (cold) evaporator and (hot) condenser. You've no doubt noticed that the A/C unit coils have fins on them: these improve thermal transfer.
So it goes with the Peltier: heatsink fins on the hot side allow it to radiate more heat. The bigger the sink, the more heat can be shed, and the more efficient the cooler can be.
What does thermal grease do, then? Without a heatsink, nothing. Its job is to improve the thermal contact to the heatsink and make it more efficient. No heatsink, no effect.
Can you run the Peltier module on 9V? That depends on the device. Can you run it on a 'transistor' 9V battery (a &quot;PPE&quot; or &quot;E&quot; type?) No, not nearly enough current.
Two more things:

Consider adding a fan to the hot-side heatsink.
Add a heatsink the the cold side too, to improve thermal transfer.

How can you make a heatsink? Extruded heatsink material is relatively easy to source, check with an electronics distributor.
You might even be able to source a suitable heatsink from scrap electronics, like an obsolete CPU motherboard or a graphics card. (Bonus: the heatsink likely would have a fan on it too.)