What would happen if we did what you're saying. Put a $1 bill in an envelope with a letter to our elected officials. Can they accept bribes by mail? Or is it only at campaign fundraisers?
I believe they need your name, address and occupation for their records to take the donation. There are exemptions for donations of less than 50 bucks but I'm not sure which circumstances they're valid in.
Email is not a good way to reach your Congressman. They get too much of it to address, and most of it is automated. Call, send a letter, or visit in person.
Don't get sucked into defeatism. The system is designed to be able to ignore the average person, sure, but here's the thing: you're not the average person.
Be smart, be motivated, and figure out how things actually work. Then apply the maximum amount of force to the place it can do the most good.
I'm not sure that place is actually phone calls, but there's a reason phone calls are suggested so often. They do work. You might be able to find something better though and if you do let us know.
Edit: Well that was dumb. I replied from memory without reading the thread again. I totally agree about email.
If you hit them where they live (personal Emil addresses and public facing forums) at least they will have to respond. These people are elected to represent the public if they aren't responding during their normal work day, we have a right to pester them until they respond.
2) Mail it to your elected pimp
3) Beg for freedom of speech