home made alcho stove, and bottle of fuel
Mora
a big sheet of Tyvec
roll of survey twine
Roll of good tape, like duck heavy duty, sticky-ass tape
boil all your water, saves you some coin.
Pots, cutlery and other cooking gear from salvation army
Fire or no fire makes a big difference with sleeping gear. In a fire-useable zone I'd try to get a wool blanket, for no fire, I'd go for as good a sleeping bag as I could find, REI or even bass-pro. it'll be heavy. An advantage with the blanket, is you could save on the backpack and use it as a gear roll.
Mat for under you, closed cell foam. camping stores, surplus stores or even anyplace that sells just rubber and foam for things. light and cheap.
Mainly the idea is stuff you can improvise with. sticking to 100$ and the clothes on your back is a bit tricky. It can be done, but your margin of error gets smaller. I don't kave prices, since I can't know what things cost where you are, but I tried to look at things that you should be able to get right now. If you took that 100$ and tolled thrift shops, bargain bins, and garage sales for six months you could probably have some pretty decent gear, but thats a lot of invested hours.