Quite Strings?

Troutki, for vinyl you unfortunately will have to search in the used market - new vinyl doesn't seem to offer much in the way of chamber music, let alone string quartets. For used vinyl, Ars Nova and Mikrokosmos are good sources to look through each month as they publish a new list:
http://home.bluemarble.net/~arsnova/
http://www.mikrokosmos.com/

The best chamber music recordings that I have are TACET DVD-A discs available from elusivedisc.com, and others.

Is there a good place to order these pieces...preferably on vinyl...but cd is ok too?

Music of string quartets is not always quiet, and can often be extremely challenging emotionally and intellectually (as in the string quartets of Shostakovich, for example). But, to answer you inquiry, some quartets that will match what you describe include the following:

J. Haydn, String Quartets from his Opus 17, 33 or 54.
Mozart, String Quartet in A, K464
Bridge, String Quartet 3
Arriaga, any of his three String Quartets
Debussy, String Quartet in g, op10
Delius, String Quartet
Wikmanson, String Quartet in e, op1, no2

Good luck in your explorations.

Not quite a string quartet, but it may be to your liking:
Respighi's "Ancient Airs and Dances"