This school is an off-shoot of the Marobashikai. The Marobashikai is headed by Watanabe Tadashige. Watanabe Tadashige's father, Tadatoshi, was a student of the 20th soke, Yagyu Toshinaga. I do not know how qualified the teachers of the Kamiizumi-kai are. To me, one yellow flag is that the teacher of neither dojo is mentioned, nor their level of attainment and/or years of training with what Sensei. They also take some potshots at the line I practice, but, well, that's budo for you.
For reference, here's
the Marobashi-kai in New York. The last kata they do is actually the old Kamiizumi-era Sangaku, so a good example of what the Kamiizumi-kai will be practicing.
the mainline, and here's
an older offshoot of the mainline for comparison.
As for legitimacy, any legitimate group should be completely happy to share their lineage. Once you've established that, the best thing I can recommend is to view various schools, or lines within schools, and find the one that impresses you most, be that the atmosphere of the dojo, the technique, or the style.
Also, you can check the home page of the Nihon Kobudo Kyokai
or contact the Nihon Kobudo Shinkokai. Membership in these organizations doesn't mean one is any good, nor does non-membership mean that something is not legitimate, but it's a place to start.