Colour Changing 10mm Diffused LED - 750mCd

If you mean these colour changing LEDs then yes https://www.kitronik.co.uk/products/components/leds/leds/colour-changing-and-rgb-leds/5v-colour-changing-led-10mm/. You can connect them directly to the USB power supply without the need for a current limiting resistor.

The LED needs to run off 5V, you are best using a 5V regulator to achieve this as you will always get 5V irrespective of the current consumption of the LED. The LED works by turning on different amounts of red, green & blue LEDs therefore to get red just the red LED is on to get purple the red and blue LED are on. This means that the current consumption of the LED when purple will be twice the current or the red as twice as many colours are on. The way you would normally work out the current limit resistor is to look at how much current the LED takes and the voltage that would drop across the resistor, the issue is that whilst there would be 4V over the resistor the current is variable. This means that you can?t work out what the value of the resistor using theory. If it is possible to keep the voltage on the LED in specification across the full colour range then you may be able to drop the voltage using a resistor however I couldn?t guarantee that it would work or recommend the resistor value.

You could use this power supply https://www.kitronik.co.uk/products/batteries-holders-chargers-usb-power/power-supplies-and-usb-power/5v-uk-mains-wall-power-supply/ without a resistor it should be able to power 100 of the colour changing LEDs. If it struggles when a lot of the LEDs change to the more power hungry colours you may wish to use 2 power supplies with 50 LEDs on each. As for a board to do this, you could try these https://www.kitronik.co.uk/products/pcb-equipment/pcbs/strip-board-pack-of-5-100x160mm/ . One last thing, the colour changing LEDs don?t stay in synch for very long, they all have slightly different timings so after a few cycles they are all showing different colours. If you want control over the colours you would need to use https://www.kitronik.co.uk/products/components/leds/leds/colour-changing-and-rgb-leds/rgb-led-5mm-common-cathode-pack-of-500/ or the common anode version and some circuitry to control the timing of the colours.

It takes about 45 seconds.