The ATF website is woefully inadequate for Title 2 weapons, you won't find much there.
As for your question, your don't need a license. In the world of firearms (and most other things) a "license" is for doing business. A Class 3 Federal Firearms License is for doing business in machine guns, suppressors, etc. If you start talking to other dealers or police about getting a license, you will be having 2 separate and confusiing conversations. You simply want to buy a machine gun, that's all. You don't need a license or a permit to do this and more than you do to buy a handgun.
To buy a machine gun, the best thing to do is locate a Class 3 FFL in your area and ask them how to go about it. Whatever you buy is going to in the form of a Form 4 transfer, meaning the specific gun you want will be transferred to you by your local dealer on an ATF Form 4. A $200 tranfer tax is paid with the submission of this form, but that is a one-time payment only. You do pay this each time you buy a new machine gun (it's sort of like Federal sales tax, helps to think of itlike that), so it's not that sort of "one-time." With the Form 4 goes passport photos, fingerprint cards and the kicker you mentioned, the CLEO signature. You do have to have your local Chief Law Enforcement Officer sign of on the Form 4 and that can be difficult, sometimes a deal breaker. This is why it makes sense to contact a local Class 3 dealer first because they will know the situation in your area. There are several "CLEOs" who can sign, including the PD Chief, Sheriff, Constable, etc. But sometimes, even frequently, none of the qualified CLEOs will sign those forms for various reasons. Without that signature, you as an individual cannot buy a Title 2 firearm. I say as an individual because you can acquire Title 2 firearms if you have a corporation, which also deletes the photo, fingerprint card and CLEO signature requirements. This has become a viable alternative for people living in areas where MGs are legal to own, but they cannot get the CLEO signatures. That's all I'll say about this because it's gets very thick.
The first step is finding the local dealer. He can either help you find exactly what it is you want or he may have it in stock ready for you. If you find the gun you want and it is out-of-state, it gets very confusing as to the transfer process as the seller, assuming he is not a dealer himself, will have to transfer the gun to his local Class 3 dealer ($200 tax paid), then that dealer will transfer it to your dealer, then your dealer to you. This whole process can and will take months. With 3-transfer deals like this, it can be 6-8 months before you get your gun. If you find someone selling what you want and he is in your state, the whole thing gets very easy as you don't need a Class 3 FFL to be involved at all. The seller can transfer the gun directly to you with the same Form 4 process. The common thread is the gun, however you find it and wherever it is located, will have to be Form 4'd to you and this single transfer will take about 3 months to approve. And transfers necessary to get it to your own Form 4 (like the seller being out of state) simply add to the time (3 transfers to approve, each occurring consecutively) and expense (possibly (2) $200 transfer taxes, multiple shipments, multiple dealer fees).
Anyway, there is a lot more to this whole thing, so it's best to consult your local dealer. They'll know all the ins and outs and can tell you the best way to go about it.