Based at Ohio State University, the Library and Museum's primary mission is to develop a comprehensive research collection documenting American printed cartoon art.  They have a great image database that is fully searchable.
The British Cartoon Archive is located in Canterbury at the University of Kent's Templeman Library.  It has a library, archive, and exhibition gallery, and is dedicated to the history of British cartooning over the last two hundred years.  The Cartoon Museum and BCA often collaborate in exhibitions.
The British Library is a major international research library that preserves the intellectual memory of the nation.  It acquired the Punch archive in 2004 though this consists mostly of documents and objects rather than pictures.
The British Museum's Prints and Drawings collection has a very extensive collection of British satirical prints and drawings from the late 18th and early 19th centuries.  It also includes some 20th century cartoons and caricatures by artists such as Ronald Searle, H.M. Bateman and Steve Bell.  The collection is searchable online and the original works can be viewed in the museum's print room. 
The UK's largest cartoonist organisation that provides a social base for UK cartoonists wherever they may live or work.
If you are interested in studying cartooning, visit this site to find the best tutors.
A dedicated home for the art of illustration in all its forms.  From adverts to animation, picture books to political cartoons and scientific drawings to fashion design. 
The Library of Congress is another great place to go in the search of British cartoons, some of which are viewable online.
The Comiclopedia is the world's largest overview of comic artists, and the brainchild of comic shop founder Kees Kousemaker.
The Cartoon Museum is part of the 'Museum Mile' which runs from King's Cross to the River Thames past thirteen of the most extraordinary museums in London.
The UK's professional cartoonists' organisation. If you are interested in commissioning cartoons, this is the place to go.
Though Punch no longer exists, they do have an online gallery of cartoons from which you can search for your favourite artist or by keyword and also pay to license or produce prints of images. 
The V&A also have a collection of cartoons that can be accessed at the National Art Library.  It is free to join for all members of the public.  
This museum in Hanover, Germany, features the world's largest collection of works by Busch, as well as contemporary comic art, illustrations and drawings.  It also houses the archive and personal collection of caricatures and cartoons of Ronald Searle.  The Museum also holds the largest collection of work by Searle in any one museum or gallery in the world.
The Trust was established to conserve and exhibit the collection of the works of William Heath Robinson made by his daughter, Joan Brinsmead.