I am currently in the midst of a project in which it would be useful to have a list of all (small) simple groups as a means to check calculations, not waste time, verify conjectures for small examples, etc.
I found this list which enumerates all groups of order $\leq 100$. This tells me that something like this is technically possible, and likely already exists, but I've not been able to find it.
Edit: Perhaps I should have mentioned this: I do not want something like the wikipedia page which has a table of the different types of simple groups. I would like something similar to the first link, which lists all groups with order $x$, then all groups of order $x+1$, etc.
I'm not sure how much more specificity I can add, but I'd be happy to answer any questions if I'm unclear.