I know almost nothing about the Rust ecosystem but am interested. What Rust packages could completely replace Curl? The top result on Google suggested hyper and reqwest.
Some likely first choices would be hyper (http client/server), rustls (encryption), tokio (async scheduling). The Rust ecosystem is quite rich in protocols and codecs, it shouldn't be too hard to find most (all ?) of the crates you need, but there's still work needed to bring them together into one curl-like tool.
Note that Rust crates tend to be more focused than what you're used to in C, made to be composed together instead of used as a one-stop-lib. So your dependency tree would look much bigger than curl's.
Hyper is a pretty robust HTTP toolkit, and reqwest is a higher-level library on top of it.