What is an organization that accepts donations?

Solution 1:

The word that is recognized by many and largely popular is charity

an organization set up to provide help and raise money for those in need.

[Oxford Dictionaries Online ]

Usage:

Choose a charity to which you want to donate.

Register your charity/charitable trust

Solution 2:

If you want to emphasize the social benefit of the organization, then charity is the right word. (You used "charitable organizations" yourself in writing this question.)

If you prefer to emphasize the organization's role in the transaction as the gift acceptor, then beneficiary would be the term to use (but only for the first sentence: "Choose the beneficiary of your donation").

Solution 3:

non-profit / nonprofit [organization]

This is more general term than charity. Examples:

I spent the next six years working for small non-profits.

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At the gathering, Obama praised a New Hampshire nonprofit that helps poor people buy fuel-efficient, reliable cars.

Solution 4:

Unlike most of the other answers here, I would avoid charity on its own. That is a specific legal status (at least in the jurisdictions I know about). You may well want to allow "Friends of Histon Road Recreation Ground" to solicit donations on your web-site, but they are not a charity, and it would be misleading to describe them as such.

Non-profit [organization] is the right answer if you are aiming at North America.

Charity or community group is probably the best phrase for the UK.