This is what I got so far in terms of address from a fellow mononofu. I never used it but hope it helps.

150-0011 Higashi, Shibuya-ku,
Tokyo 1-13-14 Shibuya Matsunaga building 1F
(Insert talent name here)

〒150-0011
東京都渋谷区東1-13-14　渋谷松永ビル1F
（株）SMOOCH内 スターダストプロモーション ○○（タレント名）係【←タレントの場合】

I don't have LINE nor hearing of management making an account for them. The only Stardust group I know that has a line is Team Syachihoko. I'll check back if I do find something about it.
But the girls do have a Twitter run by Kawakami, momowgp, and their own blogs on ameblo. They post their Twitter photos on twipple now but they put links on their Twitter to redirect you to the photo.

But for overseas fans, we have better chances at sending fanletters or coming up with videos that can catch the attention of the fandom and the girls themselves. A member here posted a birthday video for Momoka a few years back and it became popular in Japan. Last year, the same person hosted a project for MomoClo's 6th anniversary that was shown on ustream during the celebration and made everyone cry like babies in joy.
There's also their monthly Folkmura event where fans send in postcards to Fuji TV to give a message to the girls as well as song requests. Fan art is also sent there and there is a chance for the art to be displayed on the back wall. I'll supply the address when I find it.
found it~ (not sure about the full address but if you know someone who lives in Japan to send it, AWESOME!)

〒119-0188 フジテレビ
『坂崎幸之助のももいろフォーク村NEXT』

But it all depends on how you want to contact them but the nervewrecking part is the fact you don't see them receive it. But I can assure you that they will get what you send them.

But also thank other mononofus. I actually wouldn't have known the address and other contact information without them. Mononofus are so nice -w-

popularity has its price. and the price is praying for the letter to go through screenings (to ensure safety to the members), go through management and into the girls' hands.

Twitter:
https://twitter.com/momowgp

ustream:
ustream.tv/channel/momoclotv

I actually really like ustream because they archive a lot of the streams (exceptions being most performances and events). The ustreams are basically of the girls talking to their main 2 managers about events and merchandise as well as random things such as what they did and reminiscing on their adventures. You can also find their 2 24 hour ustream events (first being from 2013 to celebrate their 5th anniversary. second being 2014 for the 6th anniversary). If you're good at Japanese (or still learning), it's great practice and you get to learn more about the girls.

oh those awkward chibis.... OoO

At concerts they have baskets that you can put gifts in for each member. As an option, that might be the most direct way compared to writing letters or post cards. Also for their radio program they do read and respond to some letters on air.

MizukiChan said:

They're still trying to get back to it such as Chibi Matsuri being held in a small venue and their current AE tour all held in small venues as well. I find it nice they're going back to their old routes when the chance comes :D

That's nice, but wouldn't it be nigh impossible to attend one of their small venues? Considering it seems like even the biggest venues in Japan sell out in minutes.

They're so popular right now that they must get a lot of stuff. It's great that they read some letters live instead of just having an intern answer all of them :)) .

Ebichu also has an Official LINE account but it's less active than Syachi's. Syachi does a brilliant job with theirs, they post quite a few pics and videos.

Blog and Ustream comments aren't moderated, but the broadcast people are probably springing for someone to filter out the penis drawings.

Well, I sent the letter. Hope it reaches them.

I actually wonder how much mail they get. I know they're extremely popular, but how many people actually take the effort to write out letters in this day and age? Interesting to think about.

Food would likely be a waste of time, handlers probably won't let it near the girls for health reasons.