How does one request and receive printed Army Tech Manuals and Field Manuals?

There is usually an NCO who has the additional duty of being the publications guy. It is almost always someone in your S-1 shop. They can order TMs and FMs, but the unit has to pay for them. You are better off downloading them and printing them yourself from:

http://www.apd.army.mil/

You can find all the unclassified TMs and FMs under "Publications", then under "Doctrine and Training".

There is a similar site for the classified FMs and TMs on the SIPR, except all rules apply to them as any classified document (SIPR printer and must be stored in a GSA approved safe....and so forth).

I was a company pubs NCO many years ago, but it probably hasn't changed much.

The unit pubs guy should have an account with APD to order forms, GTAs, FMs, etc. TMs IRRC were through a separate source; supply or maintenance I think. Don't ask for regulations or DA Pams; those should be digital only.

While manuals can be purchased from the GPO, most are available on CD media that can be purchased from eBay as well as other sources. If they are for the job I recommend talking to the Section Leader or Operations NCO as they may be able to obtain them for you.