Is there a way to use a radioactive elements (Americium in my case, but for demo, any radioactive material) to create a usable amount of energy (Power small LED?) AT HOME! Using materials that can be bought from eBay/amazon and the store?
Depending on how much nuclear material you have, you can use a system based on thermoelectric materials to directly use the heat from the decay of the material to generate electricity. Following from Hazzey's comment, the safety of the experiment will quickly go out of the household realm the more power you try to generate.
Edit for OP comment 1: In theory Americium is okay to use for a thermoelectric device because it has an alpha particle decay route (easier to produce heat with alpha particles). Wikipedia says Americium based smoke detectors have 0.3 micrograms per unit so you would have to open many thousands of detectors to get a usable amount.
Attempting this as a home experiment is unlikely to be feasible.
https://en.wikipedia.org/wiki/Radioisotope_thermoelectric_generator#241Am gives the efficiency of professional prototype designs using Americum as about 2 watts / kilogram of Am. The electrical power you will get from a few micrograms of material will be undetectable (of the order of picowatts) without some very sophisticated measuring equipment, and most likely a million times too small to "power an LED" in the sense of "make the LED light up".
Take the americium as one lump ball. Surround them with a conductor with a decent (1-2 cm) air gap. Connect the conductor to one side of a capacitor, and the Americium to the the other side of the capacitor. Then connect to a circuit. Each alpha particle that decays will increase the negative charge on the Americium side, and will increase the positive charge on the conductor's side.
Once the charge has a sufficient voltage, it should discharge through the capacitor to generate the experiment. Connect to a device that emits a creepy sound, and leave in the bedroom of your worst enemy. Of course, if you really wanted measurable nuclear power at home, I'd suggest tritium