Yes, but there some qualifications, lest someone think the CMP is taking shortcuts -
1. As part of the forms you fill out, you provide info so the CMP can perform a NICS check...the same as would be done at a FFL dealer.
2. There is a statement you sign that affirms that you are eligible to purchase a rifle according to your location. (That is, you have any required permits, etc. which your state might require.)
I believe this is the reason why they are allowed to ship to your doorstep.
(There may be a little more to it, involving the original military DCM "changeover" to CMP, but the above is probably a fairly good summary.)