Technically, this is infringement
Overview of US law
In the US, Playing copyrighted music in a private setting for personal enjoyment and practice does not require any license or royalty payment. However, uploading recordings in which copyrighted music is played so that others, even a small invited group, could download copies and could, if they choose, distribute them further would be copyright infringement if done without the permission of the copyright holder. In the US such permission could be obtained with a "mechanical" license, which requires payment of a statutory fee. Or it could be requested directly from the holder or holder's agent, in which case the terms would be whatever the holder will agree to.
Making recordings of a copyrighted musical work, even if they are not distributed, also requires permission (which can be obtained through a compulsory "mechanical" license or directly).
Cites to US Copyright Law
17 USC 101 says:
To perform or display a work “publicly” means—
(1) to perform or display it at a place open to the public or at any place where a substantial number of persons outside of a normal circle of a family and its social acquaintances is gathered; or
(2) to transmit or otherwise communicate a performance or display of the work to a place specified by clause (1) or to the public, by means of any device or process, whether the members of the public capable of receiving the performance or display receive it in the same place or in separate places and at the same time or at different times.
17 USC 106 lists as among of the rights which a copyright holder has:
(1) to reproduce the copyrighted work in copies or phonorecords;
(4) in the case of literary, musical, dramatic, and choreographic works, pantomimes, and motion pictures and other audiovisual works, to perform the copyrighted work publicly;
17 USC 115 covers "mechanical" or "cover" compulsory licenses.
Practical Considerations
Given that the web site will be on an invite-only basis, it is likely that the copyright holder would never learn of it. Even if the copyright holder does learn, many holders would not consider filing suit, which would involve court fees and legal expenses, in such a case. If no fees are being charged, and the market for the copyrighted work is not harmed. the likely damages even if a suit is filed and won, are relatively small, so a suit might well not pay off.
But some holders would sue on principle, and there is no way to know in advance who would or would not. There would be an element of legal risk in putting up such a web site.
Other countries
The above is all based on united-states law. I believe that the law would be similar in most countries that have signed the Berne Copyright Convention, but the details would differ.
