Atmega32u4 has 7 VCC pins. Can I connect 1 of the 7 VCC pins to the power supply to power the MCU, and use the rest of the VCC pins (6 of them) on the MCU to power other peripherals, such as LEDs?
Not sure where you see 7. The datasheet shows 2 AVCC, 2 VCC, 1 UVCC and 1 VUSB.
The 2 AVCC are used to power the Analog circuitry, and not connecting them, and not filtering it, would mean shitty analog to digital or digital to analog conversions. If you don't need the ADC or DAC features it's not mandatory.
The VCC powers the digital circuitry. You should connect both. YMMV if you don't. Drawing too much power cab cause issues then.
The UVCC is for powering the USB circuitry. Again if you don't use it...
VBUS is actually an input that connects to USB power, for sensing when a usb cable is connected.
And there is the GND pins. All should be connected. Technically one tends to be AGND but still, connect it.
Yes you need to supply power to each Vcc pin and NO you can't feed power through the chip to other devices, LEDs etc.
The multiple VCC pins provide proper power distribution around the wafer. If you only use one a) it may not work, and b) that one pin may not be able to carry the max current and the bonding wire may burn out.
If you attempt to use the other Vcc pins as power routing, you are routing ALL the power through the chip. That just makes things worse. Further, any switching of those devices would cause current spikes in the Micro.
Addition: You ALSO need to add decoupling capacitors close to each of those Vcc pins.
Addition 2: You did not ask, but the same goes for all the ground pins.