There are many wonderful sources for full-length, legal, video of theatrical productions. Some are free. Some you have to pay for. Below are a few examples.
The Space is a project that began in the spring of 2012, as a part of the Cultural Olympiad to accompany the London Olympics. It was re-opened as a support to artists and organizations, among other things, who are exploring the use of digital transmission of their content. There are some remarkable full productions available to watch.
Library Databases (i.e. not free)
Digital Theatre Plus - provides online access to a digital streaming video collection of unique films of current, leading British theatre productions. Includes behind-the-scenes documentaries as well as teaching and learning resources to facilitate a deeper understanding of the productions and texts. Learning resources include a detailed introduction, plot summary, character biographies, a relationship map, language analysis, scene study, performance background and historical context for each play.
Films on Demand - over 25,000 high-quality digital programs and video clips, covering topics in the arts & humanities, social sciences, sciences, business and economics, and health.
Kanopy - provides a streaming platform and a broad catalog of over 26,000 streaming videos to choose from, representing more than 800 leading producers. This database includes, for example, the first 6 episodes of Shakespeare Uncovered, which is covered elsewhere on this page and about 300 of the Criterion Collection of films.
SWANK - 1,000 selected feature films and documentaries produced by major Hollywood and independent studios. Films are licensed for individual student, staff and classroom use but do not include public performance rights.
This list is not exhaustive, but it will give you a sense of what is available and where.
Digital Theatre Plus
As You Like It – RSC
As You Like it - Globe
Comedy of Errors - RSC
King Lear – Almeida
King Lear – Broadway Archive
Love's Labours Lost – Globe
Macbeth – Everyman Theatre, Liverpool
Much Ado About Nothing – West End
Much Ado About Nothing – Broadway Archive
Romeo and Juliet - Globe
Taming of the Shrew – Broadway Archive
Macbeth – From the Globe Theatre with Jeremy Brett
Macbeth – from the RSC with Antony Sher
Measure for Measure – Directed by Bob Komar
Richard II – from the Globe Theatre with David Birney
Richard III – with Laurence Olivier
The Tempest – by Derek Jarman
Kurosawa's Throne of Blood – Macbeth
Kurosawa's The Bad Sleep Well – Hamlet
Sherwood Hu's Prince of the Himalayas – Hamlet
The first season of Shakespeare Uncovered is available here, as are quite a number of other documentary type of looks at Shakespeare and his time.
As You Like It (1936)
Julius Caesar (1956) (in Turkish?)
A Midsummer Night's Dream (1909)
Le Songe d'une nuit d'ete (1969) - a psychedelic French tv version...
The Taming of the Shrew (1908) by DW Griffiths
Twelfth Night (1910)
The UWM Media Library has over 200 DVDs of Shakespeare plays & related topics. Their 'mediagraphy' is here