Re: I am going to build a true random number generator ...
Is it really reasonably available? ... You might have to buy a bunch of smoke detectors and take them apart 
Just one should be sufficient.
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Also, FWIW, it's not all that safe... It emits a bit of gamma as well, as do some of its decay products.
Well like you point out they are used unshielded in hundreds of millions of smoke detectors around the world. The risk shouldn't be anymore than adding another smoke detector to your home. My goal would be to have the source and detector sealed inside a metal box preferably one which can be mounted in a 3.5" drive bay and connect to the host via internal usb header.
1 uSv of Am-241 with no shielding produces an exposure of ~1.27 Sv per year assuming constant exposure at a distance of 1m. That is less than 1/700th of the recommended annual exposure limit of 1000 uSv annually. So "a little bit" in this case is almost zero which is why it is used in smoke detectors to begin with.
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Not too dangerous, but make sure not to eat it 
Agreed. Alpha emitter inside your body is not a good thing.