I don't remember all the details but it doesn't appear to be against regulations: https://www.ecfr.gov/cgi-bin/text-idx?SID=ca2b0d5f1c9951743ff93a43e84f3dd6&mc=true&node=sp14.2.61.c&rgn=div6#se14.2.61_183
From what I do remember, you need to get a (3rd class) medical and the medical certificate itself is your student certificate. I forget whether the instructor has to refer you to the medical examiner. I think instructors are required to conduct a background check or something. I would find it awkward to ask a CFI to do anything unless you plan on at least trying to take a lesson with them, but that's me.
BTW, since you're interested, I have been on flights and asked the pilots post-flight to get into the cockpit (at the time I was only a student) and they agreed (Jetblue A320) but El Al said (direct quote in heavy Hebrew accent) "not a chance." But it's worth asking. Also, something you might consider (maybe with a group of enthusiasts) is to get a tour of a local (Class C or D) tower. I found it interesting.
Thanks for replying.
I actually reached out to the FSDO to ask them, he told me that instructors will not do it, (he didn't want to go in to details about regulation stuff, so I guess it is not a regulation).
I mentioned above that I want to get the certificate "For increasing my odds to get permission to visit FAA facilities etc.". So I wish to visit a tower or any other ATC facility and I read on Pilot Forums that holding a certificate of any kind can make it easier to gain entry. That's what I meant.
I even called a few of them, but none of them gave permission DTC.
