I was recently in South Africa and thought I was bitten by a mosquito, but 2 days later I noticed it had turned into a ring shape. Now 2 weeks later the ring is getting a little bigger (about 3/4" diameter) and definitely itchier. Is this an insect bit or possibly ringworm? I currently do not have medical insurance and am not sure what to do

I'm not familiar with what you might get in S Africa, but it does sound like ringworm on the face of it. Ringworm is a fungus, not a worm, as you probably know, and is easily treatable. Get a $2 off coupon at Lamisil.com.
The last time I had ringworm, about 30 years ago, I remember that I was sure I had felt a really big 'bite' from an insect or something at that spot, even though it was at the top of my leg under all my clothes and while driving in my car. Maybe it was, since fungus and bacteria usually find a break in the skin, no matter how small.

Agree, if you want a non medical advice, I used to put finger nail polish on my ringworm (not a worm) it would burn, but cured it up quickly.

But many good products for it.

Keep a eye on it, if it appears infected and keep spreading, get to the doctor