Yes and no. The ps2 joystick is analog and uses 10k potentiometers for it's signals wheres the switch and vita use digital joysticks, so unfortunately it wouldn't be as easy as just hooking digital sticks up. You would need a whole lot of extra stuff to make it work.
The next best thing is the K-Silver JP19 (Forward insertion) also known as the Adafruit 5628. This is essentially a switch joystick that is converted to a analog stick and uses 10k potentiometers. Keep in mind that this is all for using the original Playstation 2 controller hardware.
If you wish to use something like the PS2+ etc. Then those have support for original switch joysticks and probably vita sticks as well.
Joycon sticks and Vita sticks output analog X and Y signals just like any other joystick. You can connect them directly to the controller, and they'll function, but the deadzones and sensitivity may not be great due to the sticks' different construction and output voltage ranges.
Vita 2000 sticks use carbon film pots (as opposed to the Vita 1000 sticks' hall effect sensors) so their ranges may be more compatible out of the box. Hard to say.