If one is looking to give Tzedakah (charity) without leaving their chair, which reliable Jewish organizations accept donations through their websites?
There are many. Pretty much every Jewish organization out there has a site with a donate button. Here are just 3 quick examples of famous ones:
Basically, if you can think of a Jewish organization, pop their name into Google (or the like) and make sure the site you're on is actually legit [not such an easy task, but common sense will generally help].
Of course there are some who don't have a donation website (at least yet). For those you'll just have to leave your seat.
Here's a list of Jewish organizations I donate online to with some regularity. I spend about 10 minutes checking out each one for reliability and quality and usefulness of work before giving a donation; feel free to do more.
With some exceptions, this list focuses on donations to the Jewish poor and needy rather than cultural institutions, mainstream educational and religious organizations, donations to non-Jews in need, or funding of large-scale projects bettering Am Yisroel or the world. Those causes, too, merit serious consideration when it comes to deciding where to give your money - please don't forget them!
The Chesed Fund collects donations for Jewish families and individuals - mostly in the U.S. and Israel - with emergency needs
You can donate through the Social and Humanitarian Programs page of Lubavitch.org to a number of Chabad-affiliated charities, including support of foster children, substance abuse treatment, disaster and terror relief, soup kitchens, and Chabad Houses/shluchim
ChesedSpot lists chesed organizations around the world for the benefit of volunteers, donors, and the needy
IsraelGives.org is an excellent, well-designed, and comprehensive registry of Israeli charities that allows you to search by category, instantly see transparency ratings for each charity, and donate online. Tzedakah gift cards are available
Jgive.com lets you quickly search among active projects and chesed organizations in Israel by type, and donate through its site
P.E.F. maintains a vetted list of recommended Israeli charities by category
Charidy.com is a (seemingly Chabad-affiliated) crowdfunding platform for Jewish and non-Jewish causes
FOR SPECIAL NEEDS/PERSONAL SUPPORT
Ohel is a Brooklyn-wide institution supporting Jews with addiction, mental health and developmental problems, family crises, trauma, and other special needs
Hasc provides programs for Jewish special-needs children in the tri-state area
Pesach Tikvah provides programs and services for Holocaust survivors, the old, and the severely disabled in Brooklyn
S.A.N.E. is a small organization in Crown Heights and Israel helping those "with nowhere else to go"
Friendship Circle provides unique social programming and connections for those with special needs, according to the Lubavitcher Rebbe's mission
RELIEF provides Jews around the world with free mental health guidance
Beineinu helps special-needs families in the U.S. and Israel
Aleh is a home in Israel for the severely disabled r"l
Ohel Sarah provides a home, support, and education for children and adults with special needs in Eretz Yisroel. After donating a small amount of money there, I received a personal letter telling me that they had used it to buy a camera and start a photography program for their residents.
FOR INVISIBLE POPULATIONS
Chabad's Aleph Institute brings the light of Yiddishkeit to incarcerated Jews and Jews at risk of incarceration. (It also cares for Jews in the U.S. Armed Forces.)
Eshel is an organization which extends kindness and concern for life to religious Jews with same-sex preference
FOR THE SICK
Hatzala, Shmira/Shomrim, Chaveirim, Shifrah u'Puah, and chevros kadisha of Brooklyn -- or ideally of your own community -- provide lifesaving care and/or dignity-promoting assistance for the urgent needs of religious Jews
Larger than Life helps Israeli children with yene machle and their parents
FOR THE POOR
Masbia soup kitchens throughout Brooklyn provide kosher food and assistance to the needy in a dignified manner
The Global Jewish Assistance and Relief Network helps Jews around the world, including the poor, the elderly, Holocaust survivors, and at-risk youth, with humanitarian aid and religious needs
Chevra Simchas Shabbos v'Yom Tov assists indigent Jews in Crown Heights with home emergencies, medical crises, and funerals (ch"v) as well as - l'havdil - Shabbos, Yom Tov, and simchas
Keren Aniyem discreetly aids Brooklyn families and individuals who cannot meet their own needs
Colel Chabad cares for the neediest people in Israel
Yad Eliezer provides food, education, medicine, religious necessities and other care to widows, orphans, soldiers, and the poor in Israel
Lema'an Achai, previously mentioned here by @DavidPearlman, gives intelligent support to the needy in Israel according to the Rambam's best practices for giving
Ten Yad offers fun, registry-style shopping to help furnish the new homes of penniless kallahs in Crown Heights
The Jewish Joint Distribution Committee builds Israel and aids needy Jews as well as victims of humanitarian disaster throughout the world
Karmey Chesed helps the poor in Israel in an attentive and hands-on way
The Rambam Charitable Trust provides free loans to Jews in need in South Africa
Healing Hearts Bank provides microloans to needy Jewish women in St. Louis, Missouri
Especially with the following organizations, please do your own research about whether your donations may be considered tzedakah and/or maaser.
Birthright helps instill a lifelong connection to Judaism and Israel in young secular Jews through free trips
A donation to Chabad's Shluchim Office provides money to families who devote literally their entire lives to being there physically, emotionally, and spiritually for the Jews in a particular community (and who receive no income or payment other than donations). Or, ideally, donate to your local Chabad House; they need it most of all!
A donation to the Halachic Organ Donor Society facilitates this spiritually and physically crucial avodah
A donation to Chabad.org helps sponsor the site's truly valuable and accessible Jewish content
Kulanu supports small, isolated Jewish communities around the world.
The Israel Free Loan Association provides free loans in Israel.
Microfy provides microloans to Jews and non-Jews in Tel Aviv.
Shamayim V'Aretz works to prevent animal suffering according to a Jewish ethos.
I realize this is not the central purpose of the question, but I feel compelled to add these notes as an option for those who may not have much cash to give.
Consider giving time by volunteering at a local chesed organization or by giving your attention and kindness to someone you know who is in need. Women can sign up as on-call volunteers to give emotional support to Jewish women with mental health issues through Chazkeinu. No effort will be wasted by signing up to learn Shas for Shidduchim, nor to say tehillim for the sick.
Clothing, household and religious items -- along with virtually anything someone else could use -- can be donated to a local gemach. Some New York gemachs are listed here
Visitors to Crown Heights can help the Aliya Institute, which supports at-risk youth in the community, by renting the Comfort Suite
You can give a direct microloan to an individual in Israel through Kiva here: https://www.kiva.org/lend?country=il
Giving tzedakah every day has made me feel substantially less helpless in the face of the suffering of the world. It has made me feel like I am a real part of the world shared by humans and by Jews, and am doing my small piece to hold it together and lift it up. It is also extremely empowering to be able to choose new directions for one's tzedakah every day, and thereby be metaken the world from many angles. It is comforting in a quite inxepressible way to be addressed by name, with love, by people who live on the street. Please join me and give something to someone now.
Colel Chabad is one. From their website:
Colel Chabad is the oldest continuously operating Tzedakah organization in Israel. It was established in 1788 by Rabbi Schneur Zalman of Liadi, founder of the Chabad Lubavitch movement. Colel Chabad is a 501(c)3 tax exempt organization.
Colel Chabad's sole purpose has always been to help the most destitute residents of Israel in a manner that preserves and enhances their dignity. For over 200 years Colel Chabad has remained faithful to its mission to help any Jew regardless of age, gender, marital status, ethnic background, location, or degree of observance.
Also, Bank of America lets me send a check directly from their website. All I do is put in how much money I want to send and 4 days later the check arrives at it's destination. This way I can send Tzedakah to any organization I want without leaving my chair or writing and delivering a check. This works great when paying for pledges given when getting an Aliyah. I am sure most other banks' online services lets you do the same thing. I can also schedule a number of checks over a period of time, which is great if I want to cut up the donation into smaller chunks.
Lema'an Achai למען אחי does great smart chessed work
I'm surpised nobody mentioned Oneg Shabbos - https://donateabox.org/ (endorsed by Rav Chaim Kanievsky)