Small board to be fitted to baseball cap?

I've recently been given my topic for my final year project as part of my degree. The project requires the use of proximity sensors and haptic feedback, all of which is to be controlled using a microcontroller and integrated onto a hat. Clearly this arrangement would need to be small so as to avoid any discomfort to the user.
My problem is that I have minimal experience with microcontrollers and the software they use. Could anyone please direct me to whatever microcontrollers would be both very small and capable of controlling ultrasonic proximity sensors with haptic feedback?
Any answers are very much appreciated! Thanks

You need to put some numbers to 'small' and work out what your ultrasonic sensor and haptic feedback will need in terms of Arduino resources. Start looking at specific parts and figure out what pins you'd need to connect them to on an Arduino. Just to give you some idea what's available I suggest you look at the Pololu Baby Orangutan, which is pretty small and has an integral h-bridge driver which you might find useful to driver your haptic motor.

This is about the smallest semi Arduino compatible I've come across

But I'm not sure it would have enough pins to both send and receive the ultrasonic stuff and control a motor.

Actually, I don't think the size of the microcontroller will be the issue. The ultrasonic transducers + electronics will be larger than the microcontroller.

Also I suspect battery size will be an issue, if you want to run the ultrasonics and also some sort of motor etc for haptic feedback for anything but short periods of time

The project requires the use of proximity sensors and haptic feedback, all of which is to be controlled using a microcontroller and integrated onto a hat. Clearly this arrangement would need to be small so as to avoid any discomfort to the user.
My problem is that I have minimal experience with microcontrollers and the software they use. Could anyone please direct me to whatever microcontrollers would be both very small and capable of controlling ultrasonic proximity sensors with haptic feedback?

Your project description is too vague. Haptic feedback doesn't tell us anything. It could be something doable or it could be totally pie in the sky. Until you specify the nature of the so called "haptic feedback" there is not much we can tell you. Can you conceal an ATtiny85 in a cap ? Sure . Can it perform haptic feedback ? (until you define that the answer remains unknown.