In order to take many FAA written exams (including most 'basic' ones such as Private Pilot Airplane, Instrument Rating Helicopter, etc.), you need to bring to the testing center a signed endorsement from an FAA-certified flight or ground instructor indicating his/her belief that you are ready to take the exam.
(note: despite such endorsements, on average about 10% of all test-takers do not pass on their first try, but we digress..)
So, to answer the question directly.. does the FAA need know or approve of how or where you receive your ground training prior to taking the written test? Well, as there is NO formal requirement for ground instruction for the written test, then the answer to that question is NO, THERE IS NO SUCH REQUIREMENT.
Now, the question is does the testing center need to know or approve where you got your training from? Again, the answer is no. The testing center is interested only in seeing an endorsement from an instructor or flight school.
Then, the question is does your instructor (or whoever does your signoff/endorsement) particularly care where you received your ground training from? Well, he/she might be interested out of general curiosity or as part of an informal process to verify that you are indeed ready to take the test, but there is certainly no official "approval" process involved here.
Let's say you use our GroundSchool program or app to study for your written test and then you need an endorsement to take your actual test at a testing center. Basically, you have two choices:
One, the PC software has a "signoff mode" where you can request that one of our instructors mails you an endorsement/signoff that you can take to a testing center to take the test. We issue such signoffs because using the software in signoff mode causes you to be rigorously tested on the material before you go take the actual test and so we are confident that this practice is both legal per the FAA and encourages better, safer pilots. So far everybody who has used our online service has passed the test on their first try.
Please note, while 'signoff mode' is available for our Windows PC versions, it is not available on all platforms.
Two, you can go to a local instructor of your choosing and have him/her give you a signoff/endorsement. Basically, to get such an endorsement, you will need to evidence to the instructor that you are ready to take the test. The amount of evidence that instructors asks for varies--for some, it can be a simple one minute conversation and in other places you may be required to sit a mock exam. In general, you can surely impress your instructor that you are ready for the test by showing him/her a few printouts from the GroundSchool software showing your impressive home-study results.