To put it in simple words -
1) Use a 3bit -7 segment decoder for the LED
2) Get the 3 bit codes for each vowel.
3) Design a counter with flip flops to sequentially count through the 3 bit codes (in the necessary order) with a clock of 5-8 secs triggering the circuit.
Re: Brain teaser: Design a circuit capable of displaying all vowels using 7 segment LCD
easiest thing is take one basic micro controller (8051) and LCD. use timers in that microcontroller. use ascii codes for generating vowels, consonents, whatever you want you can display in ascii codes.
These details given clearly in 8051 microcontrollers with assembly and C by mazidi and mazidi.