I can't recall ever seeing a 20 ga. 870 with synthetic furniture. Could be they were made, I just never saw one.
BUT the aftermarket is your friend. Speedfeed makes synthetic replacements for the 20 ga. lightweight (LW) guns, and they make a youth model set with a slightly shorter length of pull (LOP) as well. See
http://www.speedfeedinc.com/products.html for details.
Note that older 20 ga. 870s (pre- 1972 IIRC) were made on 12 ga. frames. The later models were made on 20 ga. frames, those described as lightweight. Serial numbers for full size (12 ga. size) frames for 870s end in X (2 3/4" chamber) or N (3" chamber). Lightweight or 20 ga. frame size guns have serial numbers ending in K. Replacement stocks for 12 ga guns will fit the older 20 ga. frames.
Re. the forearms, there is a law enforcement or short style forearm that's about 7" long, and doesn't come back over the front of the receiver with the action open. There's a field type or longer forearm, about 9", that does cover the front of the receiver with the action open. Field type forearms can be cut down if necessary to shorten them.
Swapping furniture on 870s is no big deal, often it does help to have the proper fore-end wrench to change out the forearm though. Sometimes impovised tools can be made to work (needlenosed pliers inserted into the notches on the forearm nut, or a tool made from flat stock or a piece of pipe).
hth,
lpl/nc