Yes, you can use your C&R (Curios & Relics) License to purchase a machine gun. There are unfortunately a lot of caveats.
The short version is that a C&R license does not allow you to skip the NFA transfer process. You still must submit fingerprints and photographs, and wait 6-9 months for your tax stamp to be processed. Once it is approved, however, a C&R eligible machine gun can be shipped interstate to you, without having to go through an NFA dealer in your state.
For the record, the ATF has a document which lists C&R eligible firearms (https://www.atf.gov/file/128116/download). In addition, any firearms 50 years old or older is also considered a Curio & Relic…which now includes every gun registered in the 1968 amnesty.
John Keene of Morphy Auctions goes into more detail in this short video with Ian.
So, if you have the money (and lots of it), you could get that Thompson, that MP40, that Lewis Gun, that fill-in-the-blank machine gun, shipped directly to your house if you have a Curios & Relics license. Oh, and are willing to wait many months and don’t live in a state that has a total ban on machine guns.