James Madison High School is regionally accredited by the Southern Association of Colleges and Schools (SACS) so it is legitimate and credits will transfer. Colleges will also accept these credits since it is regionally accredited.
These are not interactive online courses but rather correspondence courses. The student receives texts and workbooks and assignments and is, for all intents and purposes, on his own to complete the work and take exams. There is an online site to submit questions if one is having difficulty but there is no actual "teaching" done. This format is intended more for adults who have never finished high school.
Correspondence courses are the preference of many homeschoolers. However if he wants to actively participate online with other kids his own age, interactive online learning options may be more attractive. If he is self-motivated and does not procrastinate and enjoys working totally on his own and/or his mom is going to keep up with his work and make sure he completes courses in a timely manner, then this format would be a good option. There is no time limit on completing classes so if he and/or his mom don't keep up with this on a regular daily basis, then he may never graduate. Correspondence courses put the responsibility right smack on the student (and parent).
If she and her son have looked into this and have decided that this is the way he prefers to learn (self-study, no interaction with teachers or classmates) then this is a legitimate, regionally accredited high school and credits/degree will be accepted by colleges and the military.