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.
Here is a list of orders of nonabelian simple groups up to 10000. Of course, in addition, there is a abelian simple group of order each prime. You can already see from this short list that the most frequently occurring type of group is ${\rm PSL}(2,q)$ for prime powers $q$.
60: $A_5 \cong {\rm PSL}_2(4) \cong {\rm PSL}_2(5)$.
168: ${\rm PSL}_2(7) \cong {\rm PSL}_3(2)$.
360: $A_6 \cong {\rm PSL}_2(9)$.
504: ${\rm PSL}_2(8)$.
660: ${\rm PSL}_2(11)$.
1092: ${\rm PSL}_2(13)$.
2448: ${\rm PSL}_2(17)$.
2520: $A_7$.
3420: ${\rm PSL}_2(19)$.
4080: ${\rm PSL}_2(16)$.
5616: ${\rm PSL}_3(3)$.
6048: ${\rm PSU}_3(3)$.
6072: ${\rm PSL}_2(23)$.
7800: ${\rm PSL}_2(25)$.
7920: $M_{11}$.
9828: ${\rm PSL}_2(27)$.