Are there any caps that have wire leads that are in the pF range?
These definitely are available; larger pF values as ceramic, smaller as film capacitors. Need to look at a proper electronics store like Farnell/element14, mouser or digikey!

I just hope I am not looking for unicorns

No! But:

That does not require the use of PCBs (can be used without putting them on a PCB)

Uff. You have to realize everything that consists of two separate conductors can and will act as a capacitor. Therefore, things like breadboard contact rows will very much do. Their so-called parasitic capacitance will be larger than a handful pF, and the same will apply to long flying cables etc.
When building high-frequency oscillators, your mechanical way of building it becomes part of where its component values come from. I think this is really a good motivation to learn how to use at the very least perfboard to solder your own, short-wired, low-capacitance circuitry.