Thrift shops, garage sales .... buy complete junk and just limp along but at least you will be quicker than walking pace.

There are very, very few $100 bikes that will be worth buying and rideable---almost all will need further investment. Tires and tubes are almost a certainty. A rusty chain can be soaked in solvent and regreased, and while a worn chain and cassette might have your chain skipping annoyingly every time you pushed hard, the bike wouldn't be unrideable. Duct tape can fix a torn/worn seat ... The cables will probably be wrecked but as long as they aren't frozen and don't break you can use old, scratchy cables for riding around the neighborhood.

Problem is the worn bearings in the hubs and bottom bracket and pedals will likely quit at the worst possible time. The brake pads will be ossified, and the wheels likely so warped that you couldn't get the brakes to work anyway. Even if the wheels were straight enough to ride, if you didn't get them tightened up they would probably warp after a few weeks of hard riding. A spoke wrench is cheap, and there is plenty of information online ... but that spoke wrench along with all the other tools you will need eventually will add up.

If it were me I'd be looking at $25-$50 bikes and expect them to die pretty soon after buying ... but after buying a few of them (and picking up trashed bikes from dumpsters, etc for parts) I'd hopefully have enough parts to assemble a single working bike. If you are lucky they will all have the same wheel size so you can swap wheels ...which gives you three or four chances to get a good set of wheels.

Really, $100 is probably just too little money for a bike that will last a while. You might get lucky, and good if you do ... but a $25 bike will probably serve you just as well and cost just as much to be roadworthy, and you will have extra cash in case something breaks.

Once you have a pile of parts which all work together as a single reasonably reliable working bike you can start saving again. Since you won't be in a hurry you would be able to wait for a really good deal in that $100 range, and you would know enough about maintenance to be able to estimate repair costs .... and will eventually have reliable, smooth-riding bike with all parts in sufficient working order.