As mentioned in the comments, Piezas' site here has families of infinite solutions for many equations, and some complete characterizations.
There is a thread with a similar question here at math.stackexchange which may be helpful, if you remove the "online" component of your question. The answers in the thread state:
Diophantine Equations by Mordell
The Algorithmic Resolution of Diophantine Equations by Smart
Algorithms For Diophantine Equations by de Weger
History of the Theory of Numbers, Volume 2 by Dickson
EDIT: Another of the replies said Number Theory, Volume 1 and 2 by Cohen, but that he had not read either. Since someone just added it to the comments in this thread, I'm now including it here for completeness.
I would add that An Introduction to Diophantine Equations by Andreescu, designed for math olympiad training, has many equations and their solutions.
One of the answers in the math.stackexchange thread says "I imagine a complete catalog and bibliography today, bang up to date but in the same detail as Dickson, would require dozens of volumes and would be more a WIkipedia style project than anything one could reasonably contemplate publishing in book form."
This resonates with your online idea, so perhaps someone should initiate it. Certainly it would be very useful, especially to prevent wasting research time on rediscoveries.