I picked up an Ethernet shield at RadioShack and am unable to connect any components to it; does it require solder?
This one ?
Yeah.
Which Arduino do you have it mounted on? You would need a prototype board or something similar to attach components to, but that would depend on the Arduino board, not the ethernet shield.
It is mounted on a non-SMD UNO R3.
If you want to mount components, then you need a proto shield like this.
http://store.nkcelectronics.com/Shields_c_11.html
You can jumper the pins you need to a breadboard if that is not satisfactory.
So could I use an official Arduino protoshield with a tiny breadboard, and stack it under the ethernet shield, to put components through the holes on the board?
Yes, but insure the proto shield has the ICSP connector extended through the proto board, or the ethernet shield will not function. That is why I recommended that particular proto board.
Is there a pre-assembled protoshield with ICSP? I don't know how to solder...
alevinemanfail1:
Is there a pre-assembled protoshield with ICSP? I don't know how to solder...
This would be a great time to learn! You don't even have to worry about overheating any components. 
If I solder pin headers to the board, could I then simply connect the parts straight to a breadboard like on an Arduino with no shield?
If soldering is a problem for you, maybe this is a project you should try in the future after a bit of soldering practice.
Do you have the ethernet shield code working the way you want?
No, soldering is not a problem; I actually have a goal to learn it currently. I have not tested the shield itself yet.
bump
Can someone answer my question? Can I solder pin headers to the holes to allow for no soldering on the shield in the future, or do I need to solder components directly to the board?